Smarandache Function Journal, 2-3 by Don Hass

Vol. 2-3, 1993

ISSN 1053-4792

SMARANDACHE FUNCTION JOURNAL

S(n) is the smallest integer such that S(n)! is divisible by n S(n) is the smallest integer such that S(n)! is divisible by n S(n) is the smallest integer such that S(n)! is divisible by n

Number Theory Association of the

UNIVERSITY OF CRAIOVA

SFJ is periodically published in a 60-120 pages volume, and 8001000 copies. SFJ is a referred journal. SFJ is reviewed, indexed, cited by the following journals: "Zentralblatt F端r Mathematik" (Germany), "Referativnyi Zhurnal" and "Matematika" (Academia Nauk, Russia), "Mathematical Reviews" (USA), "Computing Review" (USA), "Libertas Mathematica" (USA), "Octogon" (Romania), "Indian Science Abstracts" (India), "Ulrich's International Periodicals Directory" (USA), "Gale Directory of Publications and Broadcast Media" (USA), "Historia Mathematica" (USA), "Journal of Recreational Mathematics" (USA), "The Mathematical Gazette" (U.K.), "Abstracts of Papers Presented to the American Mathematical Society" (USA), "Personal Computer World" (U.K.), "Mathematical Spectrum" (U.K.), "Bulletin of Pure and Applied Sciences" (India), Institute for Scientific Information (PA, USA), "Library of Congress Subject Headings" (USA). INDICATION TO AUTHORS Authors of papers concerning any of Smarandache type functions are encouraged to submit manuscripts to the Editors: Prof. C. Dumitrescu & Prof. V. Seleacu Department of Mathematics University of Craiova, Romania. The submitted manuscripts may be in the format of remarks, conjectures, solved/unsolved or open new proposed problems, notes, articles, miscellaneous, etc. They must be original work and camera ready [typewritten/computerized, format: 8.5 x 11 inches (. 21,6 x 28 cm)]. They are not returned, hence we advise the authors to keep a copy. The title of the paper should be writing with capital letters. The author's name has to apply in the middle of the line, near the title. References should be mentioned in the text by a number in square brackets and should be listed alphabetically. Current address followed by e-mail address should apply at the end of the paper, after the references. The paper should have at the beginning an up to a half-page abstract, followed by the key words. All manuscripts are subject to anonymous review by two or more independent sources.

A BRIEF HISTORY OF THE "SMARANDACHE FUNCTION" by Dr. Constantin Dumitrescu Department of Mathematics University of Craiova, Romania This function is originated from the exiled Romanian professor Florentin Smarandache. It is defined as follows: For any non-null ~tegers n, S(n) is the smallest integer such that (S(nÂť! is divisible by n. The importance of the notion is that it characterizes a prime number, i. e. : Let p > 4, then: p is prime i f and only i f S(p) = p. Another properties: If (a,b) = 1, then S(ab) = max ( S(a), S(b) }; and For any non-null integers, S(ab) ~ S(a) + S(b) . {All three found and proved by the author in 1979 (see [3], 15, 1213, 65).} If n > 1, then S(n) and n have a proper common divisor. {Found and proved by student Prod~nescu in 1993: as a lemma needed to solve the conjecture formulated by the author in 1979 that: the equation S(n) = S(n + 1) has no solutions (see [3], 37, and [30]).} Etc. Also, an infinity of open/unsolved problems, involving this function, provoked mathematicians around the world to study it and its applications (computational mathematics, simulation, quantum theory, etc.). Thus, the unsolved question: n

- log S (n) 1 , Calculate lim 1 + (see [3], 29) n .... oo k=2 S (k) made by the author in 1979, has been separately proved by J. Thompson from USA in 1992 (see [18], 1), by Nigel Backhouse from united Kingdom in 1993 (see [25]), and by Pal Gr0naS from Norway in 1993 (see [51]) that this limit is equal to 00. The author wonderred if it's posible to approach the function (see [3], 1979, 25-6), but Ian Parberry expressed that one can immediately find an algorithm that computes Sen) in o (nlogn/loglogn) time (see [38], 1993). Some unsolved (by now!) other problems stated by the author in 1979 (see [3], 27-30): a) To find a general form of the continued fraction expansion of S(n)/n, for all n~2. b) What is the smallest k such that for any integer n at least one of the numbers Sen), S(n+1), ... , S(n+k-1) is a

perfect square? To build the largest arithmetical progression a 1 , a 2 , ... a r for which their images by the function are also an arithmetical progression.

Etc. In 1975 Smarandache was a student at the University of Craiova, and he was attracted by the Number Theory. He created and published a lot of proposed problems of mathematics in various scientific journals. He liked to play with the numbers ... Thus, in 1980 his research paper "A Function in the Number Theory", based on a special representation of integers, was published (for the first time) in <Analele Universit~tii Timi~oara>, Seria $tiinte Matematice, Vol. 18, pp. 79-88, and was reviewed in <Zentralblatt fur Mathematik>, 471.10004, 1982, by P. Kiss, and in the <Mathematical Reviews>, 83c:10008, 1983, by R. Meyer. In 1988 he escaped from the Ceau~escu' s dictatorship, spent almost two years in a political refugee camp in Turkey (Istambul and Ankara), and finally emigrated to the United States. Articles, notes, quickies, comments, proposals related to the Smarandache Function were presented to international conferences within the Mathematical Association of America or the American Mathematical Society at the New Mexico State University (Las Cruces), New Mexico Tech. (Socorro), University of Arizona (Tucson), University of San Antonio, University of Victoria (Canada) etc. or published in <Octogon> (Sacele), <Gazeta Matematic~> (Bucharest) , <The Mathematical Spectrum> (UK) , <Elemente der Mathematik> (Switzerland), <The Fibonacci Quarterly> (USA) etc. In 1992 Dr. J. R. Sutton from United Kingdom designed a BASIC PROCedure to calculate S(n) for all powers of a prime number up to a maximum. (see [26]) Jim Duncan from United Kingdom computed up to S(1499999), the first million taking 50 hours in Lattice C on an Atari 1040ST. (see [17]) Also, John McCarthy from United Kingdom estimated that his machine would take several years to just calculate and store S(n) to disk for the entire range of n it can handle (0<n<2~32), and using the compression detailed in ncld9207.c at least 12 Gigabytes of disk space would be needed. It took about 3 hours for his program to work out that 3,303,302 pages (!) would be needed to list the full range of n and S(n). (see [15]) In 1993 Henry Ibstedt from Sweden used a dtk-computer with 486/33MHz processor in Borland's Turbo Basic and calculated S(n) for n upto 10 6 which took 2 hours and 50 minutes! (see [52]) A group of professors (V. Seleacu, C. Dumitrescu, L. Tutescu, I. P~trascu, M. Mocanu) and scientific students from the University of Craiova, having a weekly meeting, are doing research on the function and its applicability. References (chronologically) 4

[lJ Florentin Smarandache, "An Infinity of Unsolved Problems concerning a Function in the Number Theory", abstract in <Proceedings of the International Congress of Mathematicians - 1986>, Berkeley, CA, USA; [2J Constantin CordtL~eanu, abstract on this function in <Libert as Mathematica>, Texas state University, Arlington, Vol.9, 1989, 175; [3J "Smarandache Function Journal", Number Theory Publishing Co., R. Muller Editor, Phoenix, New York, Lyon, Vol.1, No.1, 1990, ISSN 1053-4792; Prof. Dr. V. Seleacu & Lect. Dr. C. Dumitrescu, Department of Mathematics, University of Craiova, Romania, editors for the next issues; registered by the Library of Congress (Washington, D. C., USA) under the code: QA .246 .S63; surveyed by <Ulrich's International Periodicals Directory> (R. R. Bowker, New Providence, NJ), 1993-94, 3437, and <The International Directory of Little Magazines and Small Presses> (Paradise, CA), 27th edition, 1991, 533; mentionned by Dr. ~erban Andronescu in <New York Spectator>, No. 39-40, March 1991, 51; reviewed by Constantin Corduneanu in <Libertas Mathematica>, tomus XI, 1991, 202; mentionned in the list of serials by <Zentralblatt fur Mathematik> (Berlin), Vol. 730, June 1992, 620; and reviewed by L. T6th (Cluj-Napoca, Romania) in <Zentralblatt fur Mathematik>, Vol. 745 (11004-11007), 1992; mentionned in <Mathematics Magazine>, Washington, D. C., Vol. 66, No.4, October 1993, 280; "Smarandache function", as a separate notion, was indexed in "Library of Congress Subject Headings", prepared by the Cataloging Policy and Support Office, Washington, D. C., 16th Edition, Vol. IV (Q-Z), 1993, 4456; [4] R. Muller, "A Conjecture about the Smarandache Function", Joint Mathematics Meetings, New Mexico State University, Las Cruces, NM, April 5, 1991; and Canadian Mathematical Society, Winter Meeting, December 9th, 1991, University of Victoria, BC; and The SouthWest Section of the Mathematical Association of America I The Arizona Mathematics Consortium, University of Arizona, Tucson, April 3, 1992; [5] Sybil P. Parker, publisher, <McGraw-Hill Dictionary of Scientific and Technical Terms>, New York, Letter to R. Muller, July 10, 1991; [6] William H. Buje, editor, <CRC Standard Mathematical Tables and Formulae>, The University of Akron, OR, Letter to R. Muller, 1991; [7] Prof. Dr. M. Razewinkel, Stichting Mathematisch Centrum I Centrum voor Wiskunde en Informatica, Amsterdam, Netherlands, Letter to R. Muller, 29 November 1991; 5

[8] Anna Hodson, Senior Editor, Cambridge University Press, England, Letter to R. Muller, 28 January 1992; [9] R. Muller, "Smarandache Function journal", note in <Small Press Review>, Paradise, CA, February 1992, Vol. 24, No. 2, p. 5; and March 1993, Vol. 25, No.3, p. 6; [10] Pilar Caravaca, editor, <vocabulario Cientifico y Tecnico>, Madrid, Spain, Letter to R. Muller, March 16, 1992; [11] Mike Mudge, "The Smarandache Function" in <Personal Computer World>, London, England, No. 112, July 1992, 420; [12] Constantin M. Popa, Conference on launching the book "America, Paradisul Diavolului / jurnal de emigrant" (Ed. Aius, dir. prof. Nicolae Marinescu, editor Ileana Petrescu, lector Florea Miu, cover by Traian Radulescu, postface by Constantin M. Popa) de Florentin Smarandache (see p. 162), Biblioteca Judeteana <Theodor Aman>, Craiova, 3 iulie 1992; [13] Florea Miu, "Interviul nostru", in <Cuvantul Libertatii>, Craiova, Romania, Anul III, Nr. 668, 14 iulie 1992, 1 & 3; [14] Mircea Moisa, "Mi,;;carea Literara Paradoxista", carnet editorial in <Cuvantul Libertatii>, Craiova, Nr. 710, 1992; [15] John McCarthy, Mansfield, Notts, U. K., "Routines for calculating S(n)" and Letter to Mike Mudge, August 12, 1992; [16] R. R. Bowker, Inc., Biography of <Florentin Smarandache>, in "American Men & Womem of Science", New Providence, NJ, 18th edition, Vol. 6 (Q-S) , 1992-3, 872; [17] Jim Duncan, Liverpool, England, "PCW Numbers Count Jully 1992 - The Smarandache Function", manuscript submitted to Mike Mudge, August 29, 1992; [18] J. Thompson, Number Theory Association, Tucson, An open problem solved (concerning the Smarandache Function) (unpublished), September 1992; [19] Thomas Martin, Proposed Problem concerning the Smarandache Function (unpublished), Phoenix, September 1992; [20] Steven Moll, editor, Grolier Inc., Danbury, CN, Letter to R. Muller, 1 October 1992; [21) Mike Mudge, "Review, July 1992 / The Smarandache Function: a first visit?" in <Personal Computer World>, London, No. 117, December 1992, 412; [22] J. Thompson, Number Theory Association, "A Property of the Smarandache Function", contributed paper, American Mathematical Society, Meeting 878, University of San Antonio, Texas, January 15, 1993; and The SouthWest Section of the Mathematical Association of America, New Mexico Tech., Socorro, NM, April 16, 1993; see "Abstracts of Papers Presented to the American Mathematical Society", Providence, RI, Issue 85, Vol. 14, No.1, 41, January 1993; 6

(23]

(24] [25]

[26] [27] [2S]

[29]

[30] [31] [32]

[33] [34] [35] [36] (37] (3S] [39]

Mike Mudge, "Mike Mudge pays a return visit to the Florentin Smarandache Function" in <Personal Computer World>, London, No. 118, February 1993, 403; David W. Sharpe, editor, <Mathematical Spectrum>, Sheffield, U.K., Letter to Th. Martin, 12 February 1993; Nigel Backhouse, Helsby, Cheshire, U. K., "Does Samma (= the Smarandache function used instead of Gamma function for sommation) exist?", Letter to Mike Mudge, February 18, 1993; Dr. J. R. Sutton, Mumbles, Swansea, U. K., "A BASIC PROCedure to calculate S(n) for all powers of a prime number" and Letter to Mike Mudge, Spring 1993; Pedro Melendez, Belo Horizonte, Brasil, Two proposed problems concerning the Smarandache Function (unpublished), May 1993; Thomas Martin, Elementary Problem B-740 (using the reverse of the Smarandache Function), in <The Fibonacci Quarterly>, Editor: Dr. Stanley Rabinowitz, Westford, MA, Vol. 31, No.2, p. lSl, May 1993; Thomas Martin, Aufgabe 1075 (using the reverse of the Smarandache Function), in <Elemente der Mathematik>, Editors: Dr. Peter Gallin & Dr. Hans Walser, CH-8494 Bauma & CH-S500 Frauenfeld, Switzerland, Vol. 48, No.3, 1993; I. Prodanescu, Problema Propusa pri vind Functia Smarandache (nepublicata), Lic. N. Balcescu, Rm. Valcea, Romania, Mai 1993; Lucian Tutescu, 0 genera1izare a Prob1emei propuse de I. Prodanescu (nepublicata), Lic. No.3, Craiova, Mai 1993; T. Pedreira, Blufton College, Ohio, "Quelques Equations Diophantiennes avec la Fonction Smarandache", abstract for the <Theorie des Nombres et Automates>, CIRM, Marseille, France, May 24-8, 1993; Prof. Dr. Bernd Wegner, editor in chief, <Zentralblatt fur Mathematik / Mathematics Abstracts>, Berlin, Letters to R. Muller, 10 July 1991, 7 June 1993; Anne Lemarchand, editrice, <Larousse>, Paris, France, Lettre vers R. Muller, 14 Juin 1993; Debra Austin, "Smarandache Function featured" in <Honeywell Pride>, Phoenix, Arizona, June 22, 1993, 8; R. Muller, "Unsolved Problems related to the Smarandache Function", Number Theory Publishing Co., Phoenix, New York, Lyon, 1993; David Dillard, soft. eng., Honeywell, Inc., Phoenix, "A question about the Smarandache Function", e-mail to <SIGACT>, July 14, 1993; Ian Parberry, Editor of <SIGACT News>, Denton, Texas, Letter to R. Muller (about computing the Smarandache Function), July 19, 1993; G. Fernandez, Paradise Valley Community College, "Smarandache Function as a Screen for the Prime Numbers" , abstract for the <Cryptography and Computational Number Theory> Conference, North Dakota State University, Fargo, ND, July 26-30, 1993; 7

[40]

[41] [42] [43] [44]

[45]

[46]

[47]

[48] [49]

(50] (51]

(52] (53] [54] [55] 8

T. Yau, "Teaching the Smarandache Function to the American Competition Students", abstract for <Mathematica Seminar>, 1993; and the American Mathematical Society Meetings, Cincinnati, Ohio, January 14, 1994; J. Rodriguez, Sonora, Mexico, Two open problems concerning the Smarandache Function (unpublished), August 1993; J. Thompson, Number Theory Association, "Some Limits involving the Smarandache Function", abstract, 1993; J. T. Yau, "Is there a Good Asymptotic Expression for the Smarandache Function", abstract, 1993; Dan Brown, Account Executive, Wolfram Research, Inc., Champaign, IL, Letter to T. Yau (about setting up the Smarandache Function on the computer using Mathematica速 software), August 17, 1993; Constantin Dumitrescu, "A Brief History of the Smarandache Function" (former version), abstract for the <Nineteenth International Congress of the History of Science>, Zaragoza, Spain, August 21-9, 1993; published in <Mathematical Spectrum>, Sheffield, UK, Vol. 26, No.2, 1993, Editor D. W. Sharpe; Florin Vasiliu, "Florentin Smarandache, Ie po~te du point sur Ie i", etude introductive au volume trilingue des po~mes haiku-s <Clopotul T~cerii / La Cloche du Silence / Silence'S Bell>, par Florentin Smarandache, Editu=a Haiku, Bucharest, translator Rodica $tef~nescu, Fall 1993, 7-8 & 121 & 150; M. Marinescu, "Nume romanesc in matematic~ ", in <Universul>, Anul IX, Nr. 199, 5, Editor Aristide Buhoiu, North Hollywood, CA, August 1993; and "Literatura paradoxist~ in-creat~ de Smarandache", in <Jurnalul de Dolj>, Director Sebastian Domozin~, Craiova, No. 38, 1-7 November, 1993; G. Vasile, "Apocalipsul ca form~ de guverna=e" , in <Baricada>, Nr.37 (192), 24, Bucharest, 14 septembrie, 1993; Mike Mudge, "Review of Numbers Count - 118 - February 1993: a revisit to The Florentin Smarandache Function", in <Personal Computer World>, London, No. 124, August 1993, 495; Pal Gr~mas, Norway, submitted theoretical results on both problems (0) & (V) from [13], to Mike Mudge, Slli~er 1993; Henry Ibstedt, Broby, Sweden, completed a great work on the problems (0) to (V), from [13], and won the <Personal Computer World>'s award (concerning some open problems related to the Smarandache Function) of August 1993; Dumitru Acu, Universitatea din Sibiu, Catedra de Matematic~, Romania, Scrisoare din 29.08.1993; Francisco Bellot Rosado, Valladolid, Spain, Letra del 02.09.1993; Dr. Petre Dini, Universite de Montreal, Quebec, e-mail du 23-Sep-1993; Ken Tauscher, Sydney, Australia, Solved problem: To find a rank for the Smarandache Function (unpublished),

[56] [57] [58] [59J [60J [61J [62 J [63J [64J [65J [66]

[67]

[68J

September, 1993; A. Stiuparu, Valcea, Problem~ propus~ rezolvat~ (unpublished), October 1993; M. Costewitz, Bordeaux, France, Generalisation du probleme 1075 de l' <Elemente der Mathematik> (unpublished), October 1993; G. Dincu (Dr~g~~ani, Romania), "Aritmogrif in Aritmetic~" / puzzle, <Abracadabra>, Anul 2, Nr. 13, 14-5, Salinas, CA, Editor Ion Bledea, November 1993; T. Yau, student, Pima Community Community College, "Alphanumerics and Solutions" (unpublished), October 1993; Dan Fornade, "Romani din Arizona", in <Luceaf~rul Romanesc>, Anul III, Nr. 35, 14, Montreal, Canada, November 1993; F. P. Mic~an, "Romani pe Mapamond", in <Europa>, Anul IV, Nr. 150, 15, 2-9 November 1993; Valentin Verzeanu, "Florentin Smarandache", in <Clipa> journal, Anaheim, CA, No. 117, 42, November 12, 1993; I. Rotaru, "Cine este F.S. ?", prefat~ la jurnalul de lag~r din Turcia "Fugit ... n, Ed. Tempus (director Gheorghe Stroe), Bucure~ti, 1993; T. Yau, student, Pima Comunity College, two proposed problems: one solved, another unsolved (unpublished), November 1993; G. Vasile, "America, America ... ", in <Acuz>, Bucharest, Anul I, No.1, 12, 8-14 November, 1993; Arizona State University, The "Florentin Smarandache Papers" Special Collection [1979 - ], processed by Carol Moore & Marilyn Wurzburger (librarian specialists), Volume: 9 linear feet, Call #: MS SC SM-15, Locn: HAYDDEN SPEC, Collections Disk 13:A:\SMARDCHE\FLRN SMA, Tempe, AZ, USA (online since November 1993); electronic mail: ICCLM@ASUACAD.BITNET, phone: (602) 965-6515; G. Fernandez, Paradise Valley Community College, "An Inequation concerning the Smarandache Function", abstract for the <Mathematical Breakthroughs in the 20th Century> Conference, State University of New York at Farmingdale, April 8-9, 1994; F. Vasiliu, "Paradoxism's Main Roots" (see "Introduction", 4), Xiquan Publ. House, Phoenix, Chicago, 1994;

The Smarandache Function, together with a sample of The Infinity of Unsolved Problems l associated with it, presented by Mike Mudge.

The Smarandache Function. Sin) (originated by Florentin Smarandache - Smarondache Function Journal. vol 1. no 1. December 1990. ISSN 10534792) is defined for all non-null integers. n. to be the smallest integer such that (S(n))! is divisible by n. Note :-.i! denotes the factorial func· tion. :-.i!=1x2x3x ... ~"i: for all positive integer :-.i. In addition O! = 1 by definition. Sin) is an even function. That is. Sin) = S(-n) since if (S(n))! is divisible by n it is also divisible by -no Sip) = P when p is a prime number. since no factorial less than p! has a factor p in this case where p is prime. The values of Sin) in Fig 1 are easily verified. For example, S(14) = 7 because 7 is the smallest number such that 7' is divisible bv 14. Problem (i) Design and implement an algorithm to generate and store/tabulate Sin) as a function of n. Hint It may be advantageous to consider the STA..."IDARD FOR.1v1 of n. viz n =ep~tp:2 ...... p:,where e = 1. p,.p, .... p, denote the distinct prime factors of n and a..a" .... a are their respective multiplicities. ' • Problem (ii) Investigate those sets of consecutive integers (i.i+ l.i+2 ... .i+x) for which S generates a monotonic increasing (or indeed monotonic decreasing) sequence. Note For (1.2.3,4.5) S generates the monotonic increasing sequence 0.2.3.4.5; here i = 1 & x = 4. If possible estimate the largest value ofx. Problem (ill) Investigate the existence of integers m.n.p.q & k with n;tm and p.rq for which: either (A): S(m) + S(m+ 1) +.... S(m+p) = Sin) + S(m+1) +.... S(n+q) or (B): Simp + S(m+1F +.... S(m+pF _k S(n)l + S(n+1)l +.... S(n+p)' Problem (iv) Find the smallest integer k

for which it is true that for all n less than some \liven no at least one of: Sin), S(n+l) .... S(n+k-1) is: :\) a perfect square B) a divisor of kn C) a factorial of a positive integer. Conjecture what happens to k as no tends to infinity: Le. becomes larger and larger. Problem (v) Construct prime numbers of the form Sin) S(n+1) ... S(n+k): where abcdefg denotes the integer formed by the concatenation of a.b.c.d.e,f & g. For example. trivially S(2) S(3) =23 which is orime. but no so triviallv S(14)S(15)S(16)S(17) = 75617. als~ prime! Definition :\n :\-SEQl)ENCE is an integer sequence a,.a, .... with 1sa,<~< ... such that no a is the sum of distinct members of th~ sequence (other than

a). Problem (vi) Investigate the construction of A-SEQUENCES at.a, .... such that the associated sequences S(a,l.S(a.,) .... are also .'\-SEQUE."iCES. DefinItion The kth order fOl"\vard finite differences of the Smarandache function are defined thus: Ds(~J = 'modulus(S(x+1) - Six)). D~kl (x) =D(D( ... k-timesDs(x) ... )) Problem (vii) Investigate the conjecture that D~kl(l) = 1 or 0 for all k greater than or equal to 2. c.f Gilbreath·s conjecture on prime numbers. discussed in 'Numbers Count' PCW Dec 1983. * Here modulus is taken to mean the absolute value of (:\13S.). modulus (y) = y if Y is positive and modulus (y) = -y if y is negative. The following selection of Diophantine Equations (i.e. solutions are sought in integer values of x) are taken from the Smarandache Jownal and make up: Problem (viii) If m & n are given integers. solve each of: a) S(x) = S(x+l). conjectured to have no

------------------------------------------------------------------1

Republished from <Personal Computer World>, No.112, 420, July . 1992 (with the author permission), because some of the follow~ng research papers are referring to these open problems. Republished from <Personal Computer World>, No.117, 1992 (with the author permission).

412, December

solution b) S(mx+n) = x c) S(mx·n) =m·nx d) S(ITL'(+n) =x, e) S(xm) =x" f) S(x)m =Six") g) S(x) • y = x - Sly). x&y not prime h) S(x) ... Sty) = S (x"'v) i) S(x.,.\') = S(x)S(\') . j) S(xyj =S(x)S(yi

Review, Julv 1992 The Smar;U;dache Function: a first visit? 2 This tooic is certain to be revisited in the ne~ future. and the lack of space available here will certainlv be remedied on that occasion. Suffice it to report that Jim Duncan computed up to S(1499999). the first million taking 50 hours in Lattice C on an Atari 1040ST. In Problem (ii). no e\'idence for a largest value of x was found, whiie in Problem (vii) the conjecture was verified for the first 32.000 values of Sin). The very worthy prizewinner is John McCarthy of 17 Mount Street. Mansfield. Notts NG19 7AT. who has extensivelv investigated the computation of S(~) up to 232 ; arriving at conclusions such as: 'several years of computing'. 'at least 12Gb of disk space' and '3.303.302 pages of output'. John's concluding comment, 'Am I mad?', is clearl\' answered NO! by examining his specimen pages of output including those relating to 10digit values of n. Listings supplied. Details from John directly upon request.

ALGORITHM IN LATTICE C TO GENERATE S (n) by Jim Duncan 9 Ryeground Lane Formby Liverpool L37 7EG ENGLAND

Computer: Atari 1040ST Run time: ca. 50 hrs to generate S(n) for 1 000 000 numbers

iinclude <stdio.h> ~include <math.h> unsigned long int pst,ast,s,t; main () {

long int n; FILE *fp; printf("input n sc~nf("Xld",

n (

2 147 483 647)\n");

~n);

fp - fopen("PRN:", KWH); fprintf(fp, "n = Xld Sen) f close (f p) ;

Xld\n", n, smaran(n».

)

(m) unsigned long int m; smar~n

unsigned long int mst,p,a,lact; double r; if

return

(0) ;

else { 1* STANDARD FORM of m *1 r = mst .. m; p = 1; fact = 1; while ( ++p <= sqrt(r» { a .. 0; while (mst X p 0) { mst = mst/p; a++; }

r = mst; if (a > 0) < pst = p; ast = a; t = S :II 0; 11

tors3(); 1* find smallest factorial 1* p~a divsor *1 i f (t > fact) fact ,. t;

(t) with *1

} }

> fact) fact .. mst; return (fact) ; if

(mst

} }

torsI () {

unsigned long int i ;

1* te~t number is pstAast *1 1* s is the difference between ~ factorial number t and ast*{pst-l) *1 1* s forms a pattern which determines the smallest value of t for which *1 1* the test number is a divisor *1 i

whi 1 e

(++i if

pst*pst 8<~, t-s (i I. pst == 0) s = s-pst+2; else s++; t += pst;

ast* (pst-I»

{

ast*<pst-1»

{

} }

tors2 () {

unsigned long int i; torsI C); i

= 0;

while (++i if

pst*pst && t-s

(i X pst == 0)

s-3*pst+4;

else s = s-2*pst+3; t +- pst; tors1(); } }

tor53 () {

unsigned long int i; tors2 ( ) ; i

= 0;

while (++i < pst*pst && t-s < ast*(pst-1» if <i I. pst == 0) s s-5*pst+6; else s = s-4*pst+5; t += pst; tors2() ; } }

{

MONOTONIC INCREASING AND DECREASING SEQUENCES OF S(n) by Jim Duncan

Pr-oblem

(11)

Monotonic incr-easing and monotonic decreasing sequences of 5(n) were investigated for x )= 6. First number in range

) x

inc 499 999 999 999 1 499 999

1 500 000 1 000 000

75 80 75

number of sequences 5 6 x = 7 x = 8 dec inc dec inc dec 83 76 63

7 14 8

10 18 10

0 1 1

inc

2 3 2

9 dec

0 1 1

Ther-e appears to be no evidence for a largest value for- x. The sequences for x = 9 are shown in Results Table 1. The existence of sequences with the same first order- finite differences was then consider-ed eg: i

i+l i+2

= = =

440 441 442

5 (i) 5Ci+l) 5 (i +2)

= = =

11 ; 14; 17;

i+l i+2

= =

5073 5074 5075

(i)

5(i+l) 5(i+2)

= = =

89 59 29

Apart fr-om the initial quartet 2,3,4,5 all such sequences with i < 1 000 000 ar-e tr-iplets. If the fir-st or-der- finite differ-ences are multiples of 6 then the 5(n) values appear to be pr-ime number-so The values are shown in Results Table 2.

RESULTS

TABLE 1

Sequences of S (n) x

n n n n n n n n n n

= = =

586951 586952 586953 586954 586955 586956 586957 586958 586959 586960

...

= = :;

= :;

n = 721970 721971 n n = 721972 n : ; 721973 n = 721974 n "" 721975 721976 n n c:: 721977 n = 721978 n =- 721979 :;

)(

= = = =

)(

Sen) Sen) Sen) Sen) Sen) Sen) Sen) Sen) Sen) Sen)

73 = 827 907 = 6067 10939 .. 28879 = 90247 :; 240659 = 360989 "" 721979

= :;

586951 73369 21739 9467 1319 1193 1181 1091 677 29

...

Sen) Sen) Sen) Sen) Sen) Sen) S(n) Sen) Sen) Sen)

Sen) Sen) Sen) Sen) Sen) Sen) Sen) Sen) Sen) Sen)

= 1473257 = 8467 = 6323 := 3877 = 3533 :; 2239 = 1999 .. 787 = 557 ::: 463

157 "" 709 1451 ::z 1607 = 6271 :; 16787 = 24799 = 303719 = 545579 = 1091159

1473257 1473258 :; 1473259 := 1473260 :; 1473261 n n := 1473262 n : ; 1473263 n == 1473264 n = 1473265 n == 1473266 n n n n

:: 9

1091150 1091151 1091152 1091153 n = 1091154 n "" 1091155 1091156 n n = 1091157 n : ; 1091158 1091159 n

n n n n

Sen) Sen) Sen) Sen) Sen) Sen) Sen) Sen) Sen) Sen)

RESULTS same same same same same same ~dme

same same same same same same same same same same same same same same same same same same same same same same same same s.ame same same same same same same same same same same same same same same same same same same same same same same same same same <;ame

difference difference difference difference difference diffe:--ence d::'路:: f e:--enc:e difference difference difference difference d iff erence difference difference difference difference difference difference difference difference difference difference difference difference difference difference difference difference difference difference difference difference difference difference difference difference difference difference difference difference difference difference difference difference difference difference difference difference difference difference difference difference difference difference di fference difference difference difference

TABLE 2

n = 4 S(n-2) = 2 S(n-l) = ~ Sen) = 4 n = 5 S(n-2) = ~ S(n-l) = 4 Sen) = 5 17 Sen) = 6 n = 18 S(n-2) = 6 S(n-l) 11 11 S(n-l) = 14 Sen) = 17 n = 442 S(n-2) 3 89 S(n-l) = 59 Sen) = 29 30 n = 5075 S(n-2) 8 Q S(n) 29 60 n = 6409 S(n-2) ~ 149 S(n-i) 127 S (n -1) == 79 S i,'l) "" ~. ~ 48 :l = 6479 S \11-::' ~.6 109 S(n-l) = 73 Sen) = 37 n = 8177 S(n-2) 84 n = 13717 S(n-2) = 211 S(n-l) = 127 S(~\ ~ ~3 379 S(n-1) 211 Sin) = 43 168 n = 20468 S(n-2) 461 S (n-l) 251 S (n) 41 210 n = 22591 S (n-2) 191 Sen) 71 = 120 n = 35145 S(n-2) 311 S(n-ll 269 Sen) 89 180 n = 59719 S(n-2) = 449 S(n-ll 401 S(n-1) 251 Sen) 101 150 n = 67771 S(n-2) 617 S(n-l) 353 Sen) 89 = 264 n = 73425 S(n-2) 74005 S(n-2) 1721 S(n-1) = 881 Sen) = 41 840 n 24 n = 82297 S(n-2) = 151 S(n-l) = 127 Sen) = 103 60 n = 104669 S(n-2) = 251 S(n-I) = 191 Sen) = 131 330 n = 111507 S(n-2) = 769 S(n-l) = 439 Sen) = 109 = 36 n = 114427 S(n-2) = 199 S(n-l) = 163 Sen) = 127 252 n = 120523 S(n-2) = 631 S(n-1) = 379 Sen) = 127 120 n = 129928 S(n-2) = 389 S(n-l) = 269 Sen) = 149 1973 S(n-1) = 1021 Sen) = 69 952 n = 146004 S(n-2) = 600 n 153520 S(n-2) 1301 S(n-l) = 701 Sen) = 101 12 n = 180482 S(n-2) = 47 S(n-l) = 59 Sen) = 71 660 n = 181485 S(n-2) = 1429 S(n-l) = 769 Sen) ~ 109 60 n = 189954 S(n-2) = 53 S(n-l) = 113 sen) = 173 = 90 n = 192067 S(n-2) = 359 S(n-l) = 269 Sen) = 179 487 Sen) 163 324 n = 198697 S(n-2) = 811 S(n-i) 839 S(n-l) 503 Sen) 167 336 n = 209752 S(n-2) 647 S (n-l) 419 S (n) 191 228 n = 227099 S (n-2) = 150 n = 231039 S(n-2) = 499 S(n-1) 349 Sen) 199 264 n = 253725 S(n-2) 727 S(n-l) 463 Sen) = 199 = 210 n = 266915 S (n-2) 631 S (n-l) 421 S (n) 211 297638 S(n-2) 1459 S(n-l) = 811 Sen) = 163 648 n = 2808 n = 306128 S(n-2) = 5669 S(n-1) = 2861 Sen) = 53 = 1320 n = 324384 S(n-2) = 2749 S(n-l) = 1429 Sen) = 109 18 n = 326163 S(n-2) = 199 S(n-l) = 191 Sen) = 163 = 240 n = 342965 S(n-2) = 719 S(n-l) = 479 Sen) = 239 = 36 n = 346390 S(n-2) = 139 S(n-1) = 103 Sen) = 67 300 n = 386906 S(n-2) = 47 S(n-l) = 347 Sen) = 647 ~ 840 n 409422 S(n-2) 1861 S(n-l) = 1021 Sen) = 181 = 270 n = 440375 S(n-2) 811 S(n-l) = 541 Sen) = 271 = 936 n = 443450 S(n-2) 2053 S(n-l) = 1117 sen) = 181 120 n 443850 S(n-2) = 509 S(n-1) = 389 Sen) = 269 443969 S(n-2) 1783 S(n-1) = 991 Sen) = 199 792 n = 450 n = 450043 S(n-2) = 1151 S(n-l) = 701 Sen) = 251 271 = 306 n = 451215 S(n-2) = 883 S(n-l) = 577 Sen) 210 n = 460559 S(n-2) 701 Sen-l) 491 Sen) = 281 464212 S(n-2) 761 S(n-1) 521 Sen) 281 240 n 991 S(n-l) 631 Sen) 271 360 n = 470727 S(n-2) 624 n 473922 S(n-2) 1481 S(n-ll = 857 Sen) = 233 90 n = 481779 Sen-2) = 449 S(n-l) = 359 Sen) = 269 = 126 n = 511688 S(n-2) 131 S(n-l) = 257 Sen) = 383 672 n = 512894 S(n-2) = 1583 S(n-l) = 911 Sen) 239 751 S (n) 271 480 n ,.. 521946 S (n-2) = 1231 S (n-l) = 714 n = 531775 S(n-2) = 1667 Sen-I) 953 Sin) 239 1693 S(n-ll 967 SIn) 241 726 n = 543455 S(n-2) 1 1

same same same same same same same same same same same same same same same same same same same same same same same same same same same

difference difference di fference difference difference difference difference difference difference difference difference difference difference difference difference difference difference difference difference difference difference difference difference difference difference d.i. fference difference

n == 565187 Sln-2) = 919 S (n-1) == 613 Sen) = 307 n 574498 S (n-2) 1381 S(n-l) == 829 sen) == 277 1176 n = 586272 S (n-2) == 2549 S (n-l) '"' 1373 Sen) :os 197 3444 n = 592537 S (n-2) :: 6971 S (n-1) = 3527 Sen) == 83 390 n 609871 S(n-2) 1091 S(n-l) == 701 Sen) == 311 840 n = 629508 S(n-2) 1931 S (n-l ) 1091 Sen) = 251 336 n == 681077 S(n-2) = 1009 S (n-l) = 673 Sin) = 337 612 n 1531 S (n-l) 919 Sen) == 307 705793 S(n-2) 78 n = 724319 S(n-2) == 467 S (n-l) == 389 Sin) == 311 264 n 726827 S (n-2) 881 S(n-1) = 617 Sen) = 353 498 n == 731179 S(n-2) == 1327 S(n-l) == S,",Q Sin) == :.31 :~(, n = 751746 S(n-::) = 339 S (n-l.l == 599 S(n) = 359 1356 n == 778837 S(n-2) 2939 S(n-l) 1583 Sen) 227 1020 n == 792675 S(n-2) 2311 S(n-1) 1291 Sen) 271 2214 n == S03427 S(n-2) 4591 S(n-l ) 2377 Sen) 163 1590 n 810451 S(n-2) == 3391 S(n-1) 1801 Sen) 211 252 n == 837969 S(n-2) == 883 S(n-l) = 631 Sen) == 379 552 n == 898783 S (n-2) = 1471 S(n-l) == 919 Sen) == 367 2850 n == 930311 S(n-2) == 5851 S(n-l) == 3001 sen) == 151 540 n == 941057 S(n-2) == 1459 S (n-l) ::; 919 Sen) == 379 240 n == 943553 S(n-2) == 881 S(n-l) == 641 Sen) == 401 120 n == 975546 S(n-2) == 619 S(n-l) == 499 Sen) == 379 2551 S (n-1) :: 1429 Sen) == 307 1122 n == 997443 S(n-2) 947 S(n-l) == 683 Sen) == 419 264 n == 1026550 S(n-2) 1028985 S(n-2) == 1747 S(n-l) == 1063 Sen) = 379 684 n 1861 S(n-l) = 1117 Sen) == 373 744 n 1042162 S(n-2) 919 Sen) == 409 510 n == 1053175 S(n-2) == 1429 S(n-l) 306

== 552

ON THE CONJ ECTU RE D. (k) (1) =1 or 0 for k>=2 by Jim Dunc an

Prob lem(v ll) For the first 32 000 S(n) 's the conje cture that Ds etc> (1) = 1 or 0 for k )= 2 is true. The ratio of the numb er of ones to the numb er of zeros appe ars to be appro xima tely 1 for large value s of k. The resu lts are shown in Resu lts Table 3. The true diffe renc es S(x-l ) - Sex} were calcu lated and the k t h order diffe renc es Ds路 etc' (I) were found to incre ase rapid ly with incre asing k. For large value s of k ( ) 100) the ratio D.s 路etc> (1)/Ds 路 e .. -1' (1) is appro xima tely equal to -2. Some value s are shown in Resu lts Table 4.

RESULTS k

Ds (k) (1)

TABLE 3 Ratio : tota l l's tota l O's

01 = 2 0100 0 = 0 0200 0 = 0 0300 0 = 0 D40(1 (l = (1 D500 0 = 0 0600 0 = 1 D700 0 = 0 0800 0 = 0 0900 0 = 0 0100 00 = 0 0110 00 = 1 0120 00 = 1 0130 00 = 1 D140 00 = 0 D150 00 = 1 0160 00 = 0 0170 00 = 0 0180 00 = 0 0190 00 = 1 0200 00 = 1 0210 00 = 0 0220 00 = 0 0230 00 = 0 0240 00 = 0 0250 00 = 1 0260 00 = 1 0270 00 = 0 0280 00 = 1 0290 00 = 0 0300 00 = 1 0310 00 = 0 0320 00 = 1

stl/s tO = 1. 112(l5 1 stl/s tO = 1.056 584 stl/s tO = 1.075 433 stl/s tO 1.015 625 stl/s tO : : 1.010 661 st1/s tO = 0.991 039 st1/s tO -- 0.990 048 st1/s tO = 0.977 014 stl/s tO = 0.987 412 st1/s tO = 0.998 201 st1/s tO = 1. 01115 4 st1/s tO = 1.010 556 st1/s tO = 1.015 036 st1/s tO = 1.018 601 st1/s tO = 1.012 748 stl/s tO = 1.004 134 stl/s tO = 1.005 308 stl/s tO = 1.004 789 stl/s tO = 1.003 903 stl/s tO = 1.004 711 stl/s tO = 1.008 129 stl/s tO = 1.004 830 st1/s tO = 1.004 620 stl/s tO = 1.003 590 stl/s tO = 1.004 571 stl/s tO = 1.001 001 stl/s tO = 1.001 260 stl/s tO = 1.004 080 stl/s tO = 1.006 018 stl/s tO = 1.005 415 stl/s tO = 1.006 408 stl/s tO = 1. 00469 9

RESULTS k

D/(k)

TABLE 4 (1)

Ds * (k)

(1)

D/(k-ll

:]951 0952 0953 D954 D9SS 0956 D957 D958 ')959

= 2.244421E+288

= =

-4. 496445t:::+288 9.009916E+288 -1.805740E+289 = 3.619671E+289 = -7. 257009E+289 = 1. 455178E+290 = -2. 918366E+290 = 5. 853595E+290 ~N60 = -1. 174247E+291 ']961 = 2.355827E+291 ::,962 = -4.726819E+291 0963 = 9. 484823E+291 0<;64 -1.903346E+292 D965 = 3.819685E+292 ~ob6 = -7. 665702E+292 0 9 67 = 1. 538452E+293 ['>962 = -3.087569E+293 0969 = 6. 196454E+293 -1.243531E+294 D970 D971 = 2. 495458E+2 9 4 D972 = -5.007457E+294 1.004734E+295 0973 0974 = -2.015791E+295 D975 = 4.043841E+295 0976 = -8. 111306E+295 D977 = 1.626784E+296 D978 = -3. 262162E+296 0979 = 6. 540525E+296 0980 = -1.311130E+297 D981 = 2. 627848E+297 D982 = -5. 265885E+297 0983 = 1.055006E+'298 0984 = -2. 113228E+298 0985 = 4.231969E+298 D986 = -8. 473041E+298 0987 = 1.696031E+299 0988 -3. 394086E+299 0989 = 6.790531E+299 D990 = -1. -35823~E+300 0991 = 2.716013E+300 0992- .- -5. 429688E+300 0993 1.085180E+301 0994 = -2. 168257E+301 1:'995 = 4.331129E+301 0996 = -8.649119E+301 0997 = 1.726721E+302 0998 = -3. 446291E+302 0999 = 6. 876396E+302 01000 = -1.371668E+303

rab 0 ratio ratio ratio r.:>_tia ratio ratio ratio ratio ratio ratio ratio ratio ratio ratio ratio ratio ratio ratia ratio rati -:J ratio ratio ratio ratio ratio ratio ratio ratio ratio ratio ratio ratio ratio ratio ratio ratio ratio ratio ratio ratio r,=lt i 0 rat i 0 r ab 0 ratio ratio rati 0 ratio ratio ratio

(1)

= -2.002974-

-2.003387

= =

-2.004170 -2.004535 -2.004881 -2.005204 -2.005504 -2.005778 -2.006026 -2. ()t)6246 -2.006437 -2.006598 -2.006728 -2.006827 -2.')06894 -2.006 9 29 -2.006932 -:. ('O69()4 -'2.006843 -2.006751 -2.006629 -2.006476 -2.006293 -2.006081 -2.005842 -2.005576 -2.005283 -2.004966 -2.004625 -2.004262 -2.003877 -2.003473 -2.003049 -2.002608 -2.002151 -2.001679 -2.001193 -2.000695 -2.000186 -1.999667 -1.999139 -1.998604 -1.998063 -1.997516 -1.996966 -1.996413 -1.995858 -1.995303 -1.994748

= -2.003787

= = = = =

= = = = = = =

= =

= = = =

= =

= = = = = = -:

= = =

A Simple Algorithm to Calculate S(n) by John C. McCarthy Introduction This short paper first outlines an "obvious" algorithm for calculating Sen) (the smallest integer m such that m! is divisible by n). Doubtless, there exist more subtle and efficient algorithms. I hope some readers will devise these and enlighten me concerning them through this journal. This is followed by a small of the algorithm.

scale investigation of the efficiency

Then there is a short discussion of a simple way of reducing the space required for storage of all Sen) for ranges of n. The storage space required for Sen) for all n which my routines can handle is considered. Heavily commented listings of an in "C", sample output and timing data th~ algorithm.

implementation of the algorithm are included to help illustrate

The Algorithm The algorithm is described in detail at the start of the header file "S(n).H". Together with "S(n).C", this forms all the code necessary to implement the algorithm. Note that, for the Sen) function to work correctly, the function make-primes() must first be called from the main program. The code for printing Sen) and timing the routines has been omitted. These activities are both implementation specific and easily done. They are therefore left as an exercise for the interested reader. The algorithm hinges on finding the prime factors of Improvements on how this is done will most benefit its efficiency. To be practical, the given works for 0<n<232. However, the any non-null integer.

implementation of the algorithm only algorithm is generally applicable to

Tables of Sen), constructed using the routines of "Sen) .e", for the largest 2000 permitted n are included. l My paging routines are rather elaborate. Using them (without printing!), it took 2.4 hours to discover that 3,745,708 pages, as tightly packed as those shown, would be required to print Sen) for all 0<n<232. Efficiency of the Algorithm In a letter to R. Muller (about Function, July 19, 1993), Ian Parberry 1

For the ~aalle~t 4800 number~, thi~ current journal.

~ee Ib~tedt'~

table (pp. 43-50) of

computing the Smarandache (editor of <SIGACT News>, 19

Denton, Texas) expressed that one can immediately find an algorithm that computes S(n) in O(nlogn/loglogn) time ('A Brief History of the "Smarandache Function"' by Dr. Constantin Dumitrescu, Department of Mathematics, University of Craiova, Romania). Disappointingly, a little analysis of the accompanying timing data on my TI85 advanced scientific calculator reveals that my algorithm is somewhat worse than this. Trying to fit the version 2 timing data to various O(f(nÂť, I obtained the following results (x=3355443200 and 10(0(x+99)-0(x-100Âť is calculated for comparison with the last entry of the version 1 timing data): O(f(nÂť

Correlation Coefficient

0(2 3Z -1) (years)

O(n) O(nlogn/loglogn) O(n/n)

0.9928879 0.9944006 0.9997756

0.6092 0.7906 24.2

10(0(x+99)-0(x-100) (milliseconds) 8909 11827 469178

O(n/n) fits the version 2 timing data best, although the time it predicts for the last entry of the version 1 timing data is almost 3 times too large. Hence, I assume the time complexity of my algorithm is a little better than O(n/n). As a rough upper limit on the

time my program (on my 20MHz 368DX for all 0<n<23Z, let us assume that every value of n requires as much time as each n in the range of the last entry of the version 1 timing data (= 159111/199/10 = 79.9553 ms). In this "worst case", it would take 10.882 years. O(n/n) time complexity predicts more than twice this value, which is a measure of how pessimistic it is. Be) would take to calculate S(n)

I would welcome a more rigorous analysis of the time complexity of my algorithm as I presently lack the necessary expertise. Simple Compression of Stored S(n) Without compression, each S(n) would be stored as a 32-bit (= 4 bytes) value. Hence 2 34 bytes (= 16 Gigabytes) would be required to store S(n) for all 0<n<2 3Z . This requirement can be reduced considerably if we use the high bit of each each byte of each value to indicate if it is the last byte of the value. If the bit is set it means that further byte(s) are required and if it is reset it means that the byte is the last byte of the current value. This means that only 7 bits of each byte are used to form the numerical part of the value. Assuming that, as with Intel format, the values are stored low-'byte' (actually 7 bits) f~rst, here are some examples: i) ii)

127 requires seven bits and so just one byte (with high bit reset to indicate no further bytes). 16,000 requires 14 bits. So it is stored as two bytes. The

first is 0 (16,000 mod 128) + 128 (to set the high bit indicating there is more to come). The second is 125 (16,000 div 128) (with high bit reset to indicate no further bytes). This reads simply as 0 (with more to follow) + 128*125 (no more to follow). iii) A number stored as the three bytes 57+128, 93+128 and 125+0 would similarly represent: 57 + 93*128 + 125*128*128 = 2,059,961. The largest numbers that can be bytes is thus as follows: 1 2 3 4 5

byte can code up to 27-1 bytes can code up to 214-1 bytes can code up to 2z1-1 bytes can code up to 2ZS-1 bytes can code up to 2 35 -1 largest unsigned long) .

= = = = =

represented

127. 16,383. 2,097,151268,435,455. 34,359,738,355 (or 8 times the

For small values of n, the savings However, even large n often have small S(n). Using this technique to ranges of n (each range was results: range of n

compression

-10,000

wi thou t with

2,147,478,648 -2,147,488,647 4,294,957,296 -4,294,967,295

by a given number of

are

considerable (400%).

compress all S(n) calculated for some also stored), I obtained the following time taken (seconds) 4.5

size

size after pkzip

4.7

40,008 15,749

19,836 15,267

without wi th

827.3 842.4

40,008 33,541

33,729 30,836

without with

1,066.2 1,085.1

40,008 34,330

34,320 31,634

The results indicate that this compression is a little better than pkzip's (a commercial file compression utility). Application of pkzip to a pre-compressed file also gives a slight improvement. Assuming that the savings shown for the middle range of 10,000 n are the average of all ranges of 10,000 n, using my compression together with that of pkzip would permit storage of S(n) for all 0<n<2 3Z in about 3.0836*23Z = 12.3344 Gigabytes. So look out for sets of 19 CD-ROMs with all your favourite numbers on them! 21st November 1993

S(n) .H

1* (c).1993.ll.l3.John.C.McCarthy "S(n) .h" Example Implementation of A Simple Algorithm to Calculate S(n), The Smarandache Function: Because there are more people familiar with C than with C++, this module has been written entirely in C (apart from "II" style comments). The module was compiled using Borland C++ version 3.1. For efficiency, n is constrained to the limits of an unsigned long. Hence, o <= n <= 2-32 - 1 (= 4,294,967,295). ("-" represents exponentiation). Although catering for n of vast magnitude is possible, it imposes heavy storage and processing overheads. The range of an unsigned long therefore seems a reasonable compromise. The algorithm depends on the most elementary properties of S(n): 1) Calculate the STANDARD FORM (SF) of n: In SF: n = +1-(pl-al)*(p2-a2)* ... *(pr-ar) where pI, p2, ... pr denote the distinct prime factors of n and aI, a2 , ... ar are their respective mu I ti p I i cit i e s . 2) S(n) = max( S(pl-al), ... ,S(pr-ar)]. 3) S(p-a), where p is prime, is given by: 3.1) a<=p ==> S(p-a) = p*a. 3.2) a>p ==> S(p-a) = x < p*a. In this case, fortunately rare, x is the smallest integer such that p appears as a factor in the list of all integers> 1 and <= x at least a times. Let the no. of times p appears as a factor in the list of all integers> 1 and <= y be f(y, pl. Then: f(y, p) = E(int(y/(p-U)] for DO while y>=(p-U. Hence, x is the smallest integer such that f(x, p»=a. Note that between succesive integer multiples of p there are no integers which have p as a factor. The trick here is to look for the largest multiple of p (call it c), such that f(p*c, p)<=a (so that x = p*c, if f(p*c, p) = a, else x = p*(c+l»: 3.2.1) c = a-2 (largest possibilty for c since f(p*(a-l), p»=a when a>p (Note: f(p*(a-l), p)=a is not sought for slight performance gain». 3.2.2) z = f(p*c, pl. 3.2.3) While(z>a): 3.2.3.1) d = no. of times p appears as a factor of p*c = (no. of times p appears as a factor of c) + 1. 3.2.3.2) c = c-l (next largest possibility for c). 3.2.3.3) z = z-d (= f(p*c, p». 3.2.4) If(z<a), x = p*(c+l). 3.2.5) Else x = p*c. To calculate the prime factors of all 32-bit n requires use only of primes < (2-16) (i.e. all primes expressible as an unsigned short integer). This is because any factor of n remaining after division of n by all its prime factors < (2-16) is simply a prime. Since there are only 6542 16-bit primes, the program first creates a list of these (which only takes about 4 seconds on my 20 MHz 386DX PC) so that they never have to be recalculated, thus saving much time.

*1 22

S (n).H #define PRIMES16 6542 II The number of 16-bit primes #define MAX_SFK 9 1* max. distinct primes in the SF of n. The smallest number with more than 9 distinct primes is the product of the 10 smallest primes (= 6,469,693,230), which is substantially more than the largest integer expressible as an unsigned long. Hence, 9 distinct primes are more than ample. *1

typedef unsigned long u_Iongi typedef unsigned int u_intj typedef enum {false, true} boolean; struct SF_struct { int sfk; u_Iong sfp(MAX_SFK] int sfa(MAX_SFK]

j j

II no. of distinct primes II the distinct primes II respective multiplicities

} j

extern u_int prime(PRIMES16+1]

II list of all 16-bit primes II plus terminating zero.

II construct list of all 16-bit primes (prime(]). II Must be called before calls to getSF() or S(). void getSF(u_long n, struct SF_struct *SF)j II calc. SF of n and store in SF u_long S(u_long n)j II calc. S(n) u_long Spa(u_long p, int ali II calc. S(p-a) where p is prime void make-primes(void)j

int f( int x, int p) j 1* the number of times the prime p appears as a factor in the integers from 1 to x inclusive. This function is only called from Spa{p, a) when a>p with x=p*(a-Z) (refer to item (3) of algorithm outline above). Max value of (a) occurs when p is a minimum, n is a maximum and {p-a)=n. So, (Z-max(a) )=max(n)=(2-3Z)-1. Hence max(a)<3Z. So, x<60 when (a) is at its max. Max value of p (and x) occurs when a=p+l and (p-a)=max(n). So, max(p)-(max(pl+ll=(Z-32)-l. The upshot is that max(p)=9 when a=10. Hence, max(x)=7Z. This explains why it is safe for x, p and the return value of f(x,p) to be passed as ints.

S (n).C 1* (c) .1993.11.13.John.C.McCarthy

"Sen) .c" Example Implementation of A Simple Algorithm to Calculate Sen), The Smarandache Function: This is the code for the module.

Refer to "S(n).h" for details.

#include "Sen) .h" u_int prime[PRIMESI6+1]

II allocate storage for list of all 16-bit primes II plus terminating zero.

void make-primes(void) {

u_int *ppj u_int *tPj u_int pj pp=primej *pp=2j *++pp=3j p=5j

II ptr to last prime so far of prime list II ptr to current test prime II number being tested for primality II II II II II

point to start of prime list set first prime to 2 set second prime to 3 next possible prime. N.B. p is kept odd so that trial division by 2 is unnecessary. whi l'e (true) { I I inf ini te loop!: tp=prime+lj II point to first odd test prime II whilst test prime <= Ip: whileÂŤ(long) *tp)*(*tp)<=p) { if(!(p%*tpÂť { II If current test prime divides (is factor of) p: p+=2; II try next odd number if(p<*pp) { II done when p overflows: *++pp=Oj II terminate list returnj tp=prime+lj else ++tPj

II point to first odd test prime II Else point to next test prime

}

II no prime <= Ip divides p so p must be prime: *++pp=p; II so store it next in the list p+=2j II try next odd number if(p<*pp) { II done when p overflows: *++pp=O; II terminate list returnj } }

S (n) . C

void QetSF(u_lonQ n, struct SF_struct *SF) {

u_int *pp; u_lonQ r;

II ptr to current prime II 'residue' of n remaininQ for factorinQ

SF->sfk=O; r=n; pp=prime;

II no. of distinct prime factors discovered II point to start of prime list

II whilst current prime <= /r and prime list not exhausted: while«(lonQ) *pp)*(*pp)<=r && *pp) { if(! (r%*pp» { II if current prime is a factor of r: SF->sfp[SF->sfkJ=*pp; II store current prime as next prime of SF SF->sfa[SF->sfkJ=l; II set its multiplicity to 1 r/=*pp; II 'divide out' current prime whi1e( !(r%*pp» { II while current prime factors r: SF->sfa(SF->sfkl++; II increment multiplicity r/=*pp; II 'divide out' current prime }

II increment count of distinct prime factors

SF->sfk++; ++PPi

if (r,> 1) { SF->sfp[SF->sfkJ=r; SF->sfa[SF->sfkJ=l; SF->sfk++;

II next prime II If II II II

n contains prime) 2-16: store it as last prime of SF set its multiplicity to 1 increment count of distinct prime factors

}

Sen) .C u_Iong Stu_long n) {

struct SF_struct SF; int sfi; u_Iong Sn; u_long X;

II II II II II

if(n==l) return 0;

II special case

ge t SF ( n, &SF);

II calc. and store SF of n

to store SF of n index of current term of SF of n current guess at Sen) S(current term of SF of n) where it might exceed current value of Sn.

II II II II

First guess at Sen) is S(p-a), where p is the largest prime in the SF of n and a is its multiplicity. This pre-empts the calculation of S(p-a) for the remaining terms where, as is likely, p*a for these terms is <= this initial guess (since S(p-a) <= p*a always): sfi=SF.sfk-1; Sn=Spa(SF.sfp[sfiJ ,SF.sfa[sfiJ); while(sfi>O) { II while more term(s): sfi"--; II next term if(SF.sfp[sfiJ*SF.sfa[sfiJ>Sn) { II if this term may have larger S(p-a): x=Spa(SF.sfp(sfiJ ,SF.sfa[sfiJ); II calc. i t if(x>Sn) Sn=x; II if new max., update Sn with it }

return Sn;

II That's all folks!

u_long Spa(u_long p, int a) {

II Refer to item 3) of the algorithm description in S(n).h. int c; II largest multiple of p such that f(p*c, p) <= a (eventually!) int Zj II f(p*c, p) int mj II used to calc. no. of times p appears as factor of c if(a<=p) return p*aj c=a-2; z=f(p*c, p) i while(z>a) { II d in items 3.2.3.1) and 3.2.3.3) of algorithm description is implicit II here: Z--i

m=C--i while(! (m%pÂť

{

II while p divides m:

Z--j

m/=pi

II 'divide out' factor of p from m

if(z<a) return P*(C+1)i else return P*Cj }

S(n) .C

int f(int x, int p) {

int k=Oi int XdPi

II count of appearance of prime p as a factor in the integers II from 1 to x. II successive divisions of x by p

xdp=x/pi whl1e(xdp)O) k+=xdpi xdp/=pi return ki }

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I'\nc::TION. S(n) . for n-4Z9.96529. &0 n-CZ9.917Z95 IJlllS: Of 4 3 1 2 0 251 .0 •••• 9 1877 11356&131 •• '77 H94966411 2939 15569 7017919 4294966427 15640al ,29.96 ... Z1 261251 15913 809 17674759 37571 cZ9.9SI.3. 9651&1 1137137 1172207 3911&27 42:94'.' •• 7 .. 2' .. " ..... 172787 202021 6567227 211427 nl61 429.96.45. t431155., 17970571 2 ••• 121 1301504'9 3067.3319 .61 7001 330&911 429.9 ••• 7. 3'.91 429.96 •• 77 429491649 3455323 17&958937 513566641 19273853 H949664as 86787 4999981 49367431 21474&3249 7895159 110&3 47721&501 41&623 1153 234417 4294916505 1259551 4,.91 124193 1951 306&71 4294968511 •• 3237 613566S47 .77211503 16777213 2251 429.9 •• 521 396, .. 7 160&'017 85Zl759 ".Z093 2147 •• 32&9 42'.'1653' 12271333 3023 25Z6450~1 16193 .. 2949 •• , .... 115191021 &4373 330312043 1129957 &910719 73&&3 42'49&6556 201547 511123 U32U 55563 347.1 4294.8&5'& 21474&329 131547309 37&93 293 1405421 OZ94~8'571 47721&51 12739 75350211 5.0"00a, 13316' ... 2949665 •• 325.7 a&S627 ... ...... 1 9717119 3.3 4294, •• 59. 1523 22253713 zg&2U57 5&&35159 15681 429496&1 •• 6551 7&4501 4294966619 105323 ,70131 42''''&6616 1Z~7573 23469763 45751 12064513 7~535419 4294966&25 5&21 3303&2049 ?a39 4294966539 .73671 4294965131 2202547 12&1697 10&0223 2&"31 ... 29.966 .... 21474.3323 a&627 139381 .121S49 17449 42949.1.57 429496 •• 5. .29.96667 21.309 152239 5564.7 ... Z949 ••• 67 42949 ••• 6. 6053 49941473 la97' IZ51771 757 4Z94966675 21401 3670911 7 23&3 3.0451517 1795. .&29.9& •••• .7119 1099&1 21&11751 165&21&3 536,70&37 4Z9496'6" 143115557 16976153 4492643 975907 429496&70& Z147U3353 431 201641&3 919 3607 119304&31 4Z~4'66711 144563 201 •• 9 137&5919 &4103981 52377.43 .294'.672. 71512779 322929&3 11164549 30460757 1020117 429.9 •• 73. 17179157 31653 14316 55513 11041 115& 11 4294"'745 47&' 191391 1299143 5811163 429496.751 18737416&9 47721853 27751 42949&6769 191 321047 4Z94966,.1 11519103 47721S531 99223 9133 1&U917 4294968771 745' 15199 34019 14&102303 14.0273 42949667.1 357U3~ 1782143' 43313503 ZI47 ... 339, 17.9 42'4966796 4294966&1 204522229 32173 761 991&2'49 4294.66 •• 1 21474&341 17&&149 359171 1515049 3257 4Z949&&lII. 350&1 1502139 4294965&29 13Z2. 29417511 429496&.2. 349753 154495Zl 1&50' 1135&6.91 4&623 4294966.31 laa7iUU 2147UHB 3m~U 7.nn 293 2909 1073741717 29~9 429496U&& 191449 26143543 9116&9 507&793 429496&171 1073741719 429.916677 3491143 1931 23633 33&213 3067&3349 42949&66&& 1135667 154423 1241363 3303620&9 429496&&96 148' 2029 176207 357913909 4Z9.96'90~ 5&39 42'49669,6 1154551 &41327 &2339 109'l • • • 9 7307 4Z9 ... 9,&91. 54,366&7 6263 572357 715127621 42949&19%7 4Z9496.,21 19522577 IS6737693 42107519 2403451 1&121 4294968,36 53923 47197439 5651%723 5990191 200&9939 4Z9.96I'.' 36497 397351 23017 353699 13&1907 429496&956 7621 97119 59&52319 226050693 70067 429496696' 9.7 26Q791 163.&7 21 ... 74&34.9 .710&.5' 4'l9496697. 9&0&1 15 •• 963 602a.7 519.51 1255.313 .. 2949669.1 .2949.7 16330673 UI07 119159 429.966997 429 .. 96699. 143165567 1597 5&35553 65519 114&3a69 4294967006 1&0007 150053 14412&41 795217 31149 429.&967011 11606019 32537629 4294967029 1&541 23917 4294967026 201 20549 &355967 1413213 Z4.5.ll 42949.7036 5009 477211561 141319 39183 18111 .. 29.96784. 21474&353 1755197 13459 25Z645121 5347 42949&705& 1431&5519 2401143 Z599659 132967 15761 .29.9.7016 107374177 1101273&1 5250571 23761 119304641 42949670" 429496709 &13566727 1132639 4294967066 Zl47413543 .29.9670&7 362 96553 76695 •• 1 1431&55699 Z1474.3549 .. 294'6709& 1252177 2609 1 .. 31655703 2135737 15&3 4294967101 236507 15053 75.297 1005&471 429.967116 63a"l 463319 1&1471 15639 21141 429496712& 2147 . . 3563 21474&357 159072857 1.&'7&63 13&547327 13147 .294967136 &5&99343 20.4249 2251031 13990121 9336&.51 .29.967146 647 5156023 1932059 21474&3579 429"67156 10737011719 1&4571 1793&3 1&01 3229 l'l12483 429491716. '49&1 62929 317159 104755297 429496717. &7"'1 143165573 5449 42949&7119 20.522247 152293 42949671" 31033 461&2443 1296007 2521 "'294967197 4294967196 25264513 "'.&Q09 1&539 113122&1 11091&9 4'l"96720' 25411 11959 523776'" 37107&1 1"'31655739 .294'672:16 13&54733 69779 10631107 22037 252645131 .29.967226 23327 70409299 2:93&9 125 .. 79 417961 42'.'67231 4111 15966421 1&77171 65543 2Ul1 .29.9672 •• 32771 .a17 82&13 21474&3629 59652323 ot2, .. '6725. 115861 35&09 295&7 37549 357531 429496726& 127 1&046061 C294967279 11&1711 3&1851 42949&7271 22605091 1039 2069 65539 715127811 4294967266

Pale 2 of 2

,,'

,,2'.'1.4"

4Z,., •• ,9.

HUUUU

23Unn

5 30199039 204522211 4294966441 2669339 19&11719 127&3

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nun

330362067 1431555627 1311 1137.3 1431655&37 3809215' 114755291 6297 9117 '47439 1431655657 42'.966911 ,,4Z7747 110127359 22453 365747 5663 10710641 215&2749 7121 &5551 017&39 21467 111967 .29 .. ,67111 102871 12&99 1163 772U .2,4967111 651&39 3513 82791 1157&731 477211579 39403369 4294967231 1431655747 16&4071 281 3U9 &951 "2''''967291

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3.9."

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7&3.1

2147.&3497 13256071 12707003 1412&151 3392549 34636&31 2&79 564533 1779191 3422&3 21691753 5197 675097 11031 1&6669 7321 70&S3 361707 21474&35a7 6197 713.497 97612&91 859&1 172&273 31&949 450&3 "2097 262657 61111&7 46614427 2147.&3647

9 57%&6219 636763 655221 5039 151"3293 133&1

29511,7 158993299 6151 2'407 7469507

C0123 56509 32"43 1&20H21 3&4 753 943'4&7 179593 2&6331107 490573 1&901 221962\ 4567 15&77 26031101 120223 5167 216331113 21&629 2111 4'l41 92.'43 .5&993349 35671 27709463 15618061 154523 7.311 317323 &1129 61291 &58993367 1579 65557 4517 2290&49 221219 9651611 95443709 6519933&3 36191 2.1a39 156993319 613 393&53 12113 41177 193 S.5&597 .5&993403 10141 85.993407 11767033 12218.9 19976591 791699 12917 395303 9199 216331141 14753 1293.7 576119 515293 151993433 7283 "4171 45210111 286331147 13109 231223 22025473 69001 4259 954.3717 17179169\ 12271335\ 65537

Timing of S(n).c Module for Calculation of Smarandache Function, version 1

Time taken to calculate S(n) depends on how easy it is to factor n. Less time is required if n has "small" prime factors. So, in the following table, the values of n shown are the mid-paints of ranges (n-99 thru n+99). Times shown are for calculating S(n) for all integers in each range 10 times over: n

time (ms)

100 200 400 800 1600 3200 6400 12800 25600 51200 10 i4 0 0 204800 409600 819200 1638400 3276800 6553600 13107200 26214400 52428800 104857600 209715200 419430400 838860800 1677721600 3355443200

268 308 345 387 432 490 571 661 766 919 2450 4036 5670 7977 10423 13004 16302 23438 29642 37011 50330 62363 77888 108179 158480 159111

Timinq of S(n).c Module for Calculation of Smarandache Function, version 2 "Time to n" is the time taken to calculate S(n) for all n <= that shown. "Time add." is the time taken to calculate S(n) for all n > previous nand <= current n. All times are in milliseconds (as per version 1): n

Time to n

Time add.

50000 100000 150000 200000 250000 300000 350000 400000 450000 500000 550000 600000 650000

18223 66763 139191 229634 335252 452539 579419 715146 859963 1012335 1171221 1336899 1508825 1686808 1870023 2058983 2252457 2450892 2653620 2860734 3072049 3288502 3509106 3733965 3962171 4194158 4429331 4668560 4910513 5155601 5404652 5656512 5911306 6169686 6431383 6696172 6963206 7232974 7505412 7779763 8056579 8336442 8620053 8905641 9194727 9486449 9780105 10076920 10375202 10676383

18223 48540 72428 90443 105618 117287 126880 135727 144816 152372 158886 165678 171927 177983 183215 188961 193473 198435 202728 207115 211314 216454 220603 224860 228206 231987 235173 239229 241953 245088 249051 251859 254794 258380 261697 264789 267034 269768 272438 274351 276816 279863 283611 285588 269086 291722 293655 296815 298282 301181

70(1000

750000 800000 850000 9QOOOO 950000 1000000 1050000 1100000 1150000 1200000 1250000 1300000 1350000 1400000 1450000 1500000 1550000 1600000 1650000 1700000 1750000 1800000 1850000 1900000 1950000 2000000 2050000 2100000 2150000 2200000 2250000 2300000 2350000 2400000 2450000 2500000

John McCarthy 17 Mount street Mansfield Notts. NG19 7AT United Kingdom

Mike Mudge pays a return visit to the Florentin Smarandache Function. l The originator of this function. Florentin Smarandache. an Eastern European mathematician. escaped from the countrY of his birth because the Communist authorities had orohibited the publication of his rese~ch papers and his participation in international congresses. After spending two years in a political refugee camp in Turkey. he emigrated to the United States. Robert~fullerofThe Number Theorv Publishing Company, PO Box 42561, Phoenix. Arizona 85080, USA, decided to publish a selection of his papers. mmencing with The Smarandache Function Journal. Vol I. No I, December 1990. ISSN 1053-4792. PCWreaders mav have met this function before, in :--iumbers Count -112July 1992, where a very encouraging response was generated. This article [February 1993) is complete in itself so don't worry if you have filed the July issue! It may be thought that those readers who attempted the previous problem-set will have an unfair advantage. However. it must be realised that no :--iumbers Count problems are completely original so previous work within a gi ven subject area is al ways a possibility and the prize is awarded using 'suitable subjective criteria' anyway, so please have a go and submit your results. however trivial they mav seem to yourself. Definition For all non-null integers. n. the Smarandache Function, Sin), is defined to be the smallest integer such that (Sin))! (The Factorial Function with argument Sin),) is di"isible by n. e.g. S(18) = 6 because 6! is divisible bv 18 but l! .... S! are not. â&#x20AC;˘ Problem (0) Design and implement an algorithm to generate and store/tabulate S(n) as a function of n upto a given

r-iiri't It may be advantageous to consider the STA""TIARD FOR.\f of n, viz n = ep~: p;, p~ ...... p~, where e=::l, and P"P,.P]' .. p, denote the distinct prime factors of n and a" a" a, ... a, are their respective multiplicities. NOTE Sin) is an even function, bv which is meant S(-n) = S(n). . Problem (i) Using either graphical or finite difference technique (Le. the construction of difference tables etc) or indeed anything else that comes to mind. address the following questions: (a) Is there a closed expression (formula) for Sin)? (b) Is there a good asymptotic expression for Sin)? (Bv which is meant a formula. which although never (in general) exact. becomes a better and better approximation to S(n) as n becomes larger and larger.) Problem (ii) For a specified non-null integer m. under what conditions does Sin) divide t.1.e difference n - m? Problem (iii) Investigate the possible integer solutions. (x.y.z) of S(x") + Sty") = S(zO) for any u greater than or equal to 1. e.g. examine the solution (5.7,2048) when u = 3. (It can be proved that an infinity of solutions exist for anv such n-value.) Compare with Fermat's Theorem reo xo+vo = zoo Problem (iv) Investigate the possibility of finding two integers n and k such that the LOGARITIDvf of S(u') to the B.-~SE S(k") is an integer. Problem (v) Recall that 'Gamma' defined as the limit as n tends to infinity of ( 1 + 1/2 + 1/3 + 1/4 .... +1/n -log(n)') exists, is known as Euler's Constant and is approximately 0.577. Investigate the possible existence of 'Samma' defined as the limit as n tends to infinity of (1 + 1IS(2) + 1/S(3) .... +1/ Sen) - !og(S(n)). Problem (vi) Find the number ofP;\RTITIONS of n as the sum of S(m) for 2<~. See PCW August 1989 and February 1990 for other problems involving PARTITIONS of n.

Republi3bed from <Personal Computer World>, No.llB, 403, February 1993 (witb tbe autbor permission), because some of tbe following research paper3 are referring to tbese open problems_

2 Republished from <Per30nal Computer World>, No.124, 1993 (witb the author permission) .

495, August

Review of '~umbers Count -118February 1993: a revisit to T:,e Florentin Smarandache Function 2 This produced a number of 'quite powerful' responses. As a note of related interest. the latest publication of FLSmarandache is 'A ~umerical Function in Congruence Theory', Libertas .Wathematica (American Romanian Academv of Arts and Science) vol 12, 1992. DO 181-185. .A.riington. Texas. â&#x20AC;˘, Pal Gronas of Norwa\, submitted theoretical results on both problems o & (v). However. the clear winner this month is a former regular respondent. now retired. Henr ... Ibstedt, Glimminge 2036. 280 60 Brobv, Sweden. Henry used a dtk-computer with 486i33MHz processor in Borland's Turbo Basic. S(n) uoto 10 6 took 2hr SOm'in. He completed a great deal of work on all problems except (vi): details of numerical results and conclusions available from Henrv or mvself to interested readers. What abo'ut problem (vi)?

by Pal Gr0naS

Enges Gate 12 7500 STa0RDAL Norway

Problem (0).

IT I17=1 p~i is the prime factorization of n, then it is easy to verify that J:

S(n) = S(IIp~i)

i=l

= max{ S(p~i) }7=lo

From this formula we see that it is essensial to determine S(pr), where p is a prime and r is a natural number. Legendres formula states that

This formula gives us a lower and an upper bound for S(pr), namely

(1)

(p - l)r + 1

S(pr)

pro

It also implies that p divides S(pr), which means that

S(pr) = p(r - i) for a particular 0

A proof of the non-existence of )) Samma:' . by Pal Grl1maS

Introduction: If TI7=1 pr' is the prime factorization of the natural number n ;::::: 2, then it is easy to verify that k

S(n) = S(IIpi') = max{ S(pii) }t:1' ;=1

From this formula we see that it is essensial to determine S(pr), where p is a prime and r is a natural number. Legendres formula states that

(1)

The definition of the Smarandache function tells us that S(pr) is the least natural number such that pr I (S(pr))!. Combining this definition with (1), it is obvious that S(pr) must satisfy the following two inequalities:

(2)

This formula (2) gives us a lower and an upper bound for S(pr), namely

(p - l)r

(3)

+1 <

S(pr) ::; pro

It also implies that p divides S(pr), which means that

S(pr)

= p(r -

i) for a particular 0 < z < [r-1 J. p

Let T(n) = 1 - log(S(n)) + 2:7:2 S(;) for n ;::::: 2. I intend to prove that liIl1n-...oo T(n) = 00, i.e. "Samma" does not exist. First of all we define the sequence PI = 2, P2 = 3, P3 = 5 and pn = the nth prime.

"Samma":

Next we consider the natural number

S(p7) 1

S(pf) m

LL-/<: i=l/<:=l S(Pi) p~

Now (3) gives us that

< Pi k ViE{l, ... ,m} and VkE{l, ... ,n} JJ. 1

Pi k

JJ. >

L/<:=2 S(k)

(4)

p~.

since S(k) > 0 for all k ~ 2, p~ ~ p~ whenever a ~ m and b ~ n and p~ = p~ if and only if a = c and b = d. Futhermore S(p~) ~ pm n, which implies that -log S(p~) ~ -log(Pm n) because log x is a strictly increasing function in the intervall [2,00). By adding this last inequality and (4), we get p~

1 -log(S(p:'))

1 S(i)

mIn 1

~ 1 -log(Pm n) + ((;

p) . ((; k)

JJ. T(Jl'mm)

JJ.

lim T(Jl'mm)

m-=

since both

I:i=l t

and

I:i=l p~

diverges as t

--+

00. In other words, liIDn_= T( n) = 00. 0

A BASIC PROCedure to calculate S(pAi) by John Sutton

16A Overland Rd. Mumbles, SWANSEA SA3 4LP, U. K.

Integer function of a single variable S$N$ S is the least integer such that 51 is divisible by N. Obviously for a prime S(p)=p include p.

since

this

the least factorial to

It is easy to see that for two primes pl>p2 S(p1*p2)=pl since this factorial is necessary to include pI and already includes p2. This generalizes to the product of any number of primes. In fact it generalizes to the product of relatively prime numbers n1 and路n2. S(nl*n2)=Max(s(nl),S(n2禄. Therefore we can simplify the general case to:

All we need now is a way of calculating S for powers of primes. start with the inverse problem: for a given factorial and a given prime what is the maximum power of the prime included? Consider p=2. All even numbers contribute a factor, all multiples of 4 contribute another, all multiples of 8 contribute yet another ... etc. So the answer is got by summing succesive DIV 2 results (DIV p in general).

Returning to the calculation of S. To do this for a single N would require factorisation of N first. A program to calculate S for all integers up to N can avoid this by doing powers of 2, then powers of 3 and their products with powers of 2 then powers of 5 etc. Calculating S for all powers of a prime up to a maximum is straightforward. A BASIC PROCedure is attached. The main program requires some care and I have not been able to finish in time.

10REM TEST PROC TOCALC S(PAI) FOR VALUES UPTO N 20: 30: 40: 50: 60INPUT"UP TO",N\ 70DIM SPP%(lOO) BODIM NPP%(lOO) 90INPUT"WHICH PR1ME",P% 100PROCSpp(P%,N%) 110FOR 1%=0 TO 100 120PR1NT SPP%(I%),NPP%(I%) 130NEXT 1% 140GOTO 60 150END 160DEF PROCSpp(P%,N%) 1701%=1 180NPP%(0)=1 1905PP%(0)=1 200J%=1 210PJ%=0 220REPEAT 230PJ%=PJ%+P% 240X%=FNinvSpp(P%,PJ%) 250REPEAT 260SPP%(1%)=PJ% 270NPP%(1%)=P%*NPP%(I%-1) 2801%=1%+1 290UNTIL 1%>X% 300J%=J%+1 310UNTIL NPP%(1%-l»N% 320ENDPROC 330DEF FNinvSpp(P%,N%) 340LOCAL S%,T% 350S%=0 360T%=N% 370REPEAT 380T%=T% D1V P% 390S%=S%+T% 400UNT1L T%<=l 410=S%

Henry Ibstedt Glirnrninge 2036 28060 Broby Sweden The Florentin Smarandache Function S(n) PCW February 1993

Problem (0)

A program SMARAND has been designed to generate S(n) up to a preset limit N (N up to 1000000 has been used in some applications). The program requires an input of prime number up to v'N. Initially the program caculates S(Pik)=D(i,k) for all primes Pi and all exponents k needed to reach the preset limit. It then proceeds to factorize consecutive values of n. If n is prime then S(n) =n otherwise Pik is replaced by D(i,k) whenever k> 1. The largest component in the resulting array is determined and is equal to S(n). Slighly different versions of the program has been used depending on the application. Up to n=32000 both S(n) and D(i,k) were registed on files. The values of S(n) for n ~ 4800 were listed with the help of a program SN_TAB and the values of S(Pik) were listed for Pi ~ 73, k ~ 75 with the help of a program SNP TAB. Problem (i)

(a) No closed expression for S(n) has been found (b) No asymptotic expression for S(n) has been found. The behaviour of S(n) for has been graphically displayed using a program SN_DISTR

~32000

Problem (ii)

S(n)/(n-m), mfO is equivalent

m=n+kS(n), k integer .... (1) Let us assume that m is a given prime p. From the definition of S(n) it is evident that for every n there exists a prime q such that q/n (or n=lq, I integer) and S(n)=jq (i integer). We can therefore write (1) in the form p=q(1 + kj) ... (2) To find solutions to (1) when m is a prime p it is therefore sufficient to chose n as a multiple of p which fills the condition 1+ kj = 1. In practice (as far as I have found) this means excluding from n those multiples of m( =p) which are divisible by primes larger than p and also cases where n-m has a different parity from S(n) as for example (n,m,S(nÂť=(5054,19,38) is not a solution while (2527,29,38) is a 38

solution. When m is not a prime let p be the largest prime such that p/m, i.e. m=rp. Solutions to (1) will then be found when n is a multipe of p for which the GCD of nand Sen) is rp. The above conditions are sufficient but may not be necessary. Lists of solutions are however easily obtained (not included) by looking for solutions to (n-m) mod Sen) = o. Problem (iii)

A number of solutions to S(xn)+S(/)=S(tt) has been obtained and listed for n=3,S,7 and 11. The program SMAR _iii uses only Smarandache function values of the type S(Pik) which had first been sorted in ascending order using a program SNP SORT. Problem (iv)

A program SMAR _iv has been designed to find solutions to the equation S(kny = S(nk) but no non-trivial solutions were found in the selected search area ~ 8000. Problem (v) In a first attempt values saved on file up to 32000 were in a program SMAR v to calculate

the sums Z(n)=1+1/S(2)+I/S(3)+ ... +1/S(n) for n= 800, 1600,2400, ...-32000. These sums were used to study the behaviour of Z(n)-T(n) for various functions T(n): T(n) =log(S(n)) gave a curve "parallel" to Zen). T(n) = log (largest prime <n) gave a similar result. T(n) = 1 + 1/2a + 1/3a + ... + l/na gave interesting results. Supplemented with a linear term a "nearly straight horizontal" line was obtained. To see if this holds for larger values the exercise was repeated for D:$1000000. Computer files to store Sen) is now out of question and the generating program SMARAND was revised so that the partial sums Zen) were calculated in the same program. T(n) was calculated in a separate program for vaious values of a. For a=O.S and it was found that 1 + 1/S(2) + 1/S(3) + ... + l/S(n) - (1 + 1/';2 + 1/';3 + ... + l/';n) - (20k-58)

. where k= n/25000 deviates from 0 with at most 10 in the interval 1 to 1000000 (at the points of representations in the graph, 1000000 was divided in 40 interval of 2S000). Problem (vi)

Not attempted. Equipment

Calculations were done on an dtk-computer with 486/33 Mhz processor. Programs were written in Borlands Turbo Basic. Printouts were done on an HP IIP Laser printer. Some graphs were done on an HP Paintjet. The run time to calculate Sen) up to 1000000 was 2 h SO m. The initial calculation up 32767 took 198 s. 39

'SMARAND, H. Ibstedt, 930320

The Smarandache function S(n) calculated by comparing largest prime and S(fY'A). The values of S(n) are calculated and registered in a file SN.OAT up to n = 32000. The calculation goes further in other applications. DEFLNG A-S CLS :T = TIMER DIM P(168},D(168,75),K(168},L(168) OPEN "PAR FOR INPUT AS #1 FOR 1=1 TO 168 :INPUT #1,P(I} :NEXT :CLOSE #1

'This part of the program calculates S(P(IY'A) and saves the result in the array O(I,A), P(I) is the Ith prime number. The routine uses the fact that D(I,A) < =P(I) *A for a downward search for the value of D(I,A). This calculation goes beyond what is required to calculate S(n) up to n = 32000. FOR 1=1 TO 42 A=2 :P=P(I} :D(I,1}=P WHILEA<76 C=O :N=O

L: C=C+1 N=N+P IF C> =A THEN D(I,A) = N :GOTO LWEND pp=p*p l1: IF N-PP*INT(NjPP)=0 THEN C=C+1 :pp=pp*p :GOTO l1 IF C> =A THEN D(I,A) = N :GOTO LWEND :ELSE L LWEND: INCRA WEND NEXT

The array O(I,A) is stored in a file SNP.OA T for future use. OPEN 路SNP.DAr FOR OUTPUT AS #2 FOR 1=1 TO 42 :FOR J=1 TO 75 PRINT #2,P(I),J,D(I,J) NEXT :NEXT :CLOSE #2

This part of the program calculates S(N). It calls on the subroutine NFACT to express N in prime factor form. Factors P(I)"A with A> 1 are replaced by O(I,A) and placed in array LO together with the factors P(I) of multiplicity 1. S(N) is then the largest component of LO. S(N) is stored in a file SN.DA T. N=1 OPEN 路SN.DAr FOR APPEND AS #3 WRITE #3,1 WHILE N <32000 INCR N :print n 'Factorize N.

GOSUB NFACT IF K(O) >0 THEN S = P(O) :GOTO LWR 'Construct LO.

FOR 1=1 TO 168 :L(I)=O :NEXT C=O FOR 1=1 TO M INCRC IF K(I) = 1 THEN L(C) = P(I) IF K(I) > 1 THEN L(C) = D(I,K(I)) NEXT 'Find the largest value of LO and hence S(N).

S=O FOR 1=1 TO C IF L(I) > S THEN S = L(I) NEXT LWR: WRITE #3,S WEND CLOSE #3 T = TIMER-T :PRINT T END 'Subroutine for factorization of N.

NFACT: FOR 1=0 TO 168 :K(I)=O :NEXT :P(O)=0 N1 =N :M=O FOR 1=1 TO 168 LA: IF N1-P(I)*INT(N1 /P(I)) =0 THEN K(I) =K(I)+ 1 :M=I :N1 =N1 /P(I) :GOTO LA IF N1 =1 THEN 1=168 NEXT IF N1 > 1 THEN P(O) = N1 :K(O) = 1 RETURN

'SN _TAB, H. Ibstedt, 930321 'This program uses the results stored in the file SN.DAT produced by the program SMAAAND to tabulate the first 4800 values of the function S(N). 'Set 1= 21 and NB = 82 on HPIIP. OEFINT I-P,S :OIM S(4800) CLS :WIOTH "LPT1:",120 :S(1)=1 :T=TIMER OPEN 路SN.OAT" FOR INPUT AS #1 FOR 1=1 TO 4800 INPUT #1 ,S(I) NEXT CLOSE #1 51$:"1 54$:111 57S=:1I1 B1$:1I 1

I I

5(n)

I" I" III

:52$:" :55$:11 : 58$:"

:B2$:"

i " :53$:"

5(n)

I" :S6S=" III :59$:11 lit :B3$:1I

5(n)

I" I" .11

11 =1 :12=75 :P1 =1 LW: LPRINT TAB(S) "The Smarandache Function S(n). LPRINTTAB(S) S1$; :FOR 1=1 TO 6 :LPRINT S2$; :NEXT :LPRINT S3$; LPRINTTAB(S) S4$; :FOR 1=1 TO 6 :LPRINT 55$; :NEXT :LPRINT S6$; LPRINT TAB(S) 57$; :FOR 1=1 TO 6 :LPRINT 58$; :NEXT :LPRINT 59$; FOR 1=11 TO 12 LPRINT TAB(S) "nO; FOR J=O TO 7 LPRINT USING "#####";1 +J*75; :LPRINT路 ""; :LPRINT USING "#####";S(I +J*75); LPRINT" 0" ; NEXT NEXT LPRINT TAB(S) B1$; :FOR 1=1 TO 6 :LPRINT 82$; :NEXT :LPRINT B3$; LPRINT TA8(S) "Page"P1 "of S. LPRINT CHR$(12) IF P1 =1 THEN P1 =2 :11 =601 :12=675 :GOTO LW IF P1 =2 THEN P1 =3 :11 =1201 :12=1275 :GOTO LW IF P1 =3 THEN P1 =4 :11 =1801 :12=1S75 :GOTO LW IF P1 =4 THEN P1 =5 :11 =2401 :12=2475 :GOTO LW IF P1 =5 THEN P1 =6 :11 =3001 :12=3075 :GOTO LW IF P1 =6 THEN P1 =7 :11 =3601 :12=3675 :GOTO LW IF P1 = 7 THEN P1 =S :11 =4201 :12=4275 :GOTO LW PRINT "ENO" :ENO

The Smerendache Function Sen).

n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23

24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62

63 64 65 66 67 68

Sen)

2 3 4 5 3 7 4 6 5

II 4 13 7 5 6 17 6 19 5 7

II 23 4 10 13 9 7 29 5 31 8 11 17 7 6 37 19 13 5 41 7 43

II 6 23 47 6 14 10 17 13 53 9 11 7 19 29 59 5 61 31 7 8 13

II 67 17 23 7

69 70 71

73 74 75

6 73 37 10

Page 1 of 8.

n 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150

Sen) 19 11 13 79 6 9 41 83 7 17 43 29 11 89 6 13 23 31 47 19 8 97 14 11 10 101 17 103 13 7 53 107 9 109 11 37 7 113 19 23 29 13 59 17 5 22 61 41 31

Sen)

151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171

151 19 17 11 31 13 157 79 53 8 23 9 163 41

173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199

zoo

7 127 8 43 13 131 11 19 67 9 17 137 23 139 7 47 71 13 6 29 73 14 37 149 10

201 202 203 204 205 206 207

II 83 167 7 26 17 19 43 173 29 10

II 59 89 179 6 181 13 61 23 37 31 17 47 9 19 191 8 193 97 13 14 197

II 199 10 67 101 29 17 41 103 23

210 211 212 213 214 215 216 217 218 219

13 19 7 211 53 71 107 43 9 31 109 73

2ZO

221 222 223 224

17 37

Z08 209

225

n 226

227 228 229 230 231 232 233 234

235 236

237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271

273 274 275 276 277 278 279 280 281 282

288

289 290

291 292 293 294 295 296

300

Sen)

376 377 378 379

47 29 9 379 19 127 191

41 113 151 227 13 19 457

451 452 453 454 455 456 457 458 459

460

461

388

193 43 97

389

390 391 392 393 394 395 396 397 398 399

13 23 14 131 197 79

465

526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567

293

368

14 59 37 11 149 23 10

369 370

371

446

447

373 374 375

31 373 17 15

303 304

305

29 233 13 47 59 79 17 239 6 241 22 12 61 14 41 19 31 83 15 251 7 23 127 17 10 257 43 37 13 29 131 263 11 53 19 89 67

307

269

344

9 271 17 13 137

345

II 23 277 139 31 7 281 47 71 19 13 41 8 34 29 97 73

287

301 302

306

283

Z86

Sen)

284

285

43 151 101 19 61 17 307 11 103 31 311 13 313 157 7 79 317 53 29 8 107 23 19 9 13 163 109 41 47 11 331 83 37 167 67 7 337 26 113 17 31 19 21 43 23 173 347 29 349 10 13 11 353 59 71 89 17 179 359 6 38 181 22 13 73 61 367 23 41 37

113 227 19 229 23

283

297 298 299

223

Sen)

308 309

310 311 312 313 314 315 316 317 318 319 320 321 322 323

324 325 326

327 328

329 330

331 332

334

335 336

337 338

339 340

341 342 343 346

347 348 349 350 351 352 353 354 355 356 357 358 359 360

361 362 363 364

365 366

367

380

381 382 383 384 385 386

387

400

401 402 ~

404 405

406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444

445 448

449 450

383

462

17 23 461 11

463

464

29 31 233 467 13 67 47 157 59 43 79 19 17 53 239 479 8 37 241 23 22 97 12 487 61 163 14 491 41 29 19

466

467 468

469 470 471

397 199 19 10 401 67 31 101 9 29 37 17

409

484

41 137 103 59 23 83

13 139 19 419 7 421 211 47 53 17 71 61 107 13 43 431 9 433 31 29 109 23 73 439

II 14 17 443 37 89 223 149 8 449 10

229

473 474 475 476 477 478 479 480

481 482

48J 48S 486

487 488 489

490 491 492 493 494 495 496 497 498 499 500 501 502 503

504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525

II 31 71 83 499 15 167 251 503 7 101 23 26 127 509 17 73 12 19 257 103 43 47 37 173 13 521 29 523 131 10

568

569 570 571 5n 573 574 575 576 577 578 579 580 581 582 583 584 585 586 587 588 589 590 591 592 593 594 595 596 597 598 599 600

Sen) 263 31 11 46 53 59 19 41 89 107 67 179 269 14 9 541 271 181 17 109 13 547 137 61

II 29 23

79 277 37 139 557 31 43 7 17 281 563 47 113 283 9 71 569 19 571 13 191 41 23 8 577 34 193 29 83

97 53 73 13 293 587 14 31 59 197 37 593

II 17 149 199 23 599 10

The s..randache Function S(n).

, n

S(n)

601

60Z 603 604

676 6n 678 679

624

67 151 22 101 607 19 29 61 47 17 613 307 41 11 617 103 619 31 23 311 89 13

62S

700

626

313 19 157 37 7 631 79 211 317 127 53 14 29 71 8 641 107 643 23 43 19 647 9 59 13 31 163 653 109 131 41 73 47 659 11 661 331 17 83 19 37 29 167 223 67 61 8 673 337 10

701

605 606

607 608 609

610 611 612 613 614 615 616 617 618 619 6ZO

621 6Z2 623

627 62! 629

630

631 632 633 634

635 636

637 638

639 640 641 642 643 644

645 646

647 648

649 650 651 652 653 654

655 656

657 658 659 660

661 66Z 663 664

665 666

667 668 669

670 671 6n 673 674 675

Page 2 of 8.

680

681 682 683 684

685 686

687 688

689 690

691 692

693 694 695 696

697 698 699

702 703 704

705 706

707 708 709

S(n)

26 6n 113 97 17 227 31 683

19 137 21 229 43 53 23 691 173 11 347 139 29 41 349

709

734

367 14 23 67 41 739 37 19 53 743 31 149 373 83 17 107 15

7Z6

n7 n8 729

735 736

737 738

739 740 741 742 743 744 745 746 747 748 749 750

751 752 753

751 47 251 29 151 9 757 379 23 19 761 127 109 191 17

826 827

59 827 23

901 90Z

754

755 756 757 758 759 760 761 762 763 764 765 766 767 768 769 770 771

m m

775

733

S(n)

10 701 13 37 11 47 353 101 59

733

nl n2

730 731 732

720

S(n)

Z33

71 79 89 31 17 13 179 239 359 719 6 103 38 241 181 29 22 n7 13 15 73 43 61

710 711 712 713 714 715 716 717 718 719

778 779 780

781 782 783 784 785 786

787 788 789 790

791

793 794

795 796

797 798 799 800

SOl 80Z 803

804 805 806

S07 808 809

810 8" 812 813 814 815 816 817 818 819

383

59 10 769 11 257 193

43 31 97 37

389

41 13 71 23 29 14 157 131 787 197 263 79 113 11 61 397 53 199 797 19 47 10 89 401 73 67 23 31 269 101 809

9 811 29 271 37 163 17

8Z8 8Z9 830

910 911 912 913 914 915 916 917 918 919

845

920

846

47 22 53 283 17 37 71 853 61 19 107 857 13 859 43 41 431

921

833 834

835 836

837 838

839 840

841 842 843 844

847 848

849 850 851 852 853 854

855 856

857 858 859 860

861 862 863 864

865 866

867 868

869 870 871 8n 873 874 875 876 8n 878 879 880

881 88Z 883 884 885 886

887 888

889 890

891 892 893 894 895 896

8Z2

409 13 41 821 137

823

898

824

103 11

899 900

820

8Z5

903 904 905 906

83 2n 13 17 139 167 19 31 419 839 7 58 421 281 211

831 832

821

8Z9

897

907 908 909

922 9Z3

924 925 9Z6 927

53 41 43 113 181 151 907 227 101 13 911 19 83 457 61

463

929

31 19

932

Z33

933 934

311 467 17 13 937 67 313 47 941 157 41 59 9 43 947 79 73 19 317 17 953 53 191 239 29 479 137 8 62 37 107 241 193 23 967 22 19 97 971 12 139 487 13

937 938

9 173 433 34 31 79 29 67 109 97 23 15

939

951 952 953

940

941 942

943 944

945 946

947 948

949 950

293

954

11 881 14

955 956

883

958

17 59 443 887 37 127 89 11 223 47 149 179 8 23 449 31 10

959

957 960

961 962 963 964 965 966

967 968 969

970 971 9n 973 974 975

S(n)

976 9n 978 979

61 9n 163 89 14 109 491 983 41 197 29 47 19 43 11 991 31 331 71 199 83 997 499 37 15 13 167 59 251 67 503 53 7 1009 101 337 23 1013

1051 1052 lOS3 1054 lOSS 1056 lOS7 lOS8 1059 1060 1061 1062 1063 1064 1065 1066 1067 1068 1069 1070 1071 10n 1073 1074 1075 1076 10n 1078 1079 1080 1081 1082 1083 1084 1085 1086 1087 1088 1089 1090 1091 1092 1093 1094 1095 1096 1097 1098 1099 1100 1101 1102 1103 1104 1lOS 1106 1107 1108 1109 1110 ,',1 1112 ,',3 1114 1115 1116 1117 1118 1119 1120 1121 1122 1123 1124 1125

1051 263 13 31 211 11 151 46 353 53 1061 59 1063 19 71 41 97 89 1069 107 17 67 37 179 43 269 359 14 83 9 47 541 38 271 31 181 1087 17 22 109 1091 13 1093 547 73 137 1097 61 157 11 367 29 1103 23 17 79 41 2n 1109 37 101 139 53 557 223 31 1117 43 373 8 59 17 1123 281 15

1126 1127 1128 1129 1130 1131 1132 1133 1134 1135 1136 1137 1138 1139 1140 1141 1142 1143 1144 1145 1146 1147 1148 1149 1150 1151 1152 1153 1154 1155 1156 1157 1158 1159 1160 1161 1162 1163 1164 1165 1166 1167 1168 1169 1170 1171 11n 1173 1174 1175 1176 11n 1178 1179 l1SO 1181 1182 1183 1184 1185 1186 1187 1188 1189 1190 1191 1192 1193 1194 1195 1196 1197 1198 1199 1200

563 23 47 1129 113 29

980

981 98Z

983 984

985 986

987 988 989 990

992 993 994 995 996

929

991

930

936

S(n)

131 17 919 23 307 461 71 11 37 103 29

931

229

92S

863

8n 439

I S(n)

997 998 999

1000 1001 1002 1003 1004 1005 1006 1007 1008 1009 1010 1011 1012 1013 1014 1015 1016 1017 1018 1019 1020 1021 1022 1023 1024 1025 1026 1027 1028 1029 1030 lOll 1032 1033 1034 1035 1036 1037 1038 1039 1040 1041 1042 1043 1044 1045 1046 1047 1048 1049 1OS0

29 127 113 509 1019 17 1021 73 31 12 41 19 79 257 21 103 1031 43 1033 47 23 37 61 173 1039 13 347 521 149 29 19 523 349 131 1049 10

Z83

103 9 227 71 379 569 67 19 163 571 127 13 229

191 37 41 383

23 1151 8 1153 5n 11 34 89 193 61 29 43 83 1163 97 Z33

53 389

73 167 13 1171 293

23 587 47 14 107 31 131 59 1181 197 Z6

37 79 593 1187

41 17 397 149 1193 199 239 23 19 599 109 10

The s.,.andadle Function S(n). S(n)

S(n)

1201 1202 1203 1204 1205 1206 1207 1208 1209 1210 1211 1212 1213 1214 1215 1216 1217 1218 1219 1220 1221 1222 1223 1224 1225 1226 1227 1228 1229 1230 1231 1232 1233 1234 1235 1236 1237 1238 1239 1240 1241 1242 1243 1244 1245 1246 1247 1248 1249 1250 1251 1252 1253 1254 1255 1256 1257 1258 1259

1201 601 401 43 241 67 71 151 31 22 173 101 1213 607 12 19 1217 29 53 61 37 47 1223 17 14 613

1276 1277 1278 1279 1280 1281 1282 1283 1284 1285 1286 1287 1288 1289 1290 1291 1292 1293 1294 1295 1296 1297 1298 1299 1300 1301 1302 1303 1304 1305 1306 1307 1308 1309 1310 1311 1312 1313 1314 1315 1316 1317 1318 1319 1320 1321 1322 1323 1324 1325 1326 1327 1328 1329 1330 1331 1332 1m 1334 1335 1336 1337 1338 1339 1340 1341 1342 1343 1344 1345 1346 1347 1348 1349 1350

29 1277 71 1279 10 61 641 1283 107 257 643 13 23 1289 43 1291 19 431 647 37 9 1297 59 433 13 1301 31 1303 163 29 653 1307 109 17 131 23 41 101 73

1351 1352 1353 1354 1355 1356 1357 1358 1359 1360 1361 1362 1363 1364 1365 1366 1367 1368 1369 1370 1371 1372 1373 1374 1375 1376 1377 1378 1379 1380 1381 1382 1383 1384 1385 1386 1387 1388 1389 1390 1391 1392 1393 1394 1395 1396 1397 1398 1399 1400 1401 1402 1403 1404 1405 1406 1407 1408 1409 1410 1411 1412 1413 1414 1415 1416 1417 1418 1419 1420 1421 1422 1423 1424 1425

193 26 41 677 271 113 59 97 151 17 1361 227 47 31 13 683 1367 19 74 137 457 21 1373 229 15 43 17 53 197 23 1381 691 461 173 277 11 73 347 463 139 107 29 199 41 31 349 127 233 1399 10 467 701 61 13 281 37 67 11 1409 47 83 353 157 101 283 59 109 709 43 71 29 79 1423 89 19

1426 1427 1428 1429 1430 1431 1432 1433 1434 1435 1436 1437 1438 1439 1440 1441 1442 1443 1444 1445 1446 1447 1448 1449 1450 1451 1452 1453 1454 1455 1456 1457 1458 1459 1460 1461 1462 1463 1464 1465 1466 1467 1468 1469 1470 1471 1472 1473 1474 1475 1476 1477 1478 1479 1480 1481 1482 1483 1484 1485 1486 1487 1488 1489 1490 1491 1492 1493 1494 1495 1496 1497 1498 1499 1500

31 1427 17 1429 13 53 179 1433 239 41 359 479 719 1439 8 131 103 37 38 34 241 1447 181 23 29 1451 22 1453 727 97 13 47 15 1459 73 487 43 19 61 293 733 163 367 113 14 1471 23 491 67 59 41 211 739 29 37 1481 19 1483 53 11 743 1487 31 1489 149 71 373 1493 83 23 17 499 107 1499 15

1501 1502 1503 1504 1505 1506 1507 1508 1509 1510 1511 1512 1513 1514 1515 1516 1517 1518 1519 1520 1521 1522 1523 1524 1525 1526 1527 1528 1529 1530 1531 1532 1533 1534 1535 1536 1537 1538 1539 1540 1541 1542 1543 1544 1545 1546 1547 1548 1549 1550 1551 1552 1553 1554 1555 1556 1557 1558 1559

79 751 167 47 43 251 137 29 503 151 1511 9 89 757 101 379 41 23 31 19 26 761 1523 127 61 109 509 191 139 17 1531 383 73 59 307 12 53 769 19 11 67 257 1543 193 103 m 17 43 1549 31 47 97 1553 37 311 389 173 41 1559 13 223 71 521 23 313 29 1567 14 523 157 1571 131 22 787 10

1576 1577 1578 1579 1sao 1581 1582 1583 1584 1585 1586 1587 1588 1589 1590 1591 1592 1593 1594 1595 1596 1597 1598 1599 1600

1579 79 31 113 1583 11 317 61 46 397 227 53 43 199 59 797 29 19 1597 47 41 10

1601

1602 1603 1604 1605 1606

89 229 401 107 73

1607

1608 1609 1610 1611 1612 1613 1614 1615 1616 1617 1618 1619 1620 1621 1622 1623 1624 1625 1626 1627 1628 1629 1630 1631 1632 1633 1634 1635 1636 1637 1638 1639 1640 1641 1642 1643 1644 1645 1646 1647 1648 1649 1650

67 1609 23 179 31 1613 269 19 101 14

1260

1261 1262 1263 1264 1265 1266 1267 1268 1269 1270 1271 1272 1273 1274 1275

409

307 1229 41 1231 11 137 617 19 103 1237 619 59 31 73 23 113 311 83 89 43 13 1249 20 139 313 179 19 251 157 419 37 1259 7 97 631 421 79 23 211 181 317 47 127 41 53 67 14 17

Page 3 of 8.

263

47 439 659 1319 11 1321 661 14 331 53 17 1327 83 443 19 33 37 43 29 89 167 191 223 103 67 149 61 79 8 269 673 449 337 71 10

1560

1561 1562 1563 1564 1565 1566 1567 1568 1569 1570 1571 1572 1573 1574 1575

197 83 263

809

1619 9 1621 811 541 29 15 271 1627 37 181 163 233 17 71 43 109 409

1637 13 149 41 547 821 53 137 47 823

61 103 97 11

S(n)

1651 1652 1653 1654 1655 1656 1657 1658 1659 1660 1661 1662 1663 1664 1665 1666 1667 1668 1669 1670 1671 1672 1673 1674 1675 1676 1677 1678 1679 1680 1681 1682 1683 1684 1685 1686 1687 1688 1689 1690 1691 1692 1693 1694 1695 1696 1697 1698 1699 1700 1701 1702 1703 1704 1705 1706 1707 1708 1709 1710 1711 1712 1713 1714 1715 1716 1717 1718 1719 1720 1721 1722 1723 1724 1725

127 59 29 827 331 23 1657

1726 1727 1728 1729 1730 1731 1732 1733 1734 1735 1736 1737 1738 1739 1740 1741 1742 1743 1744 1745 1746 1747 1748 1749 1750 1751 1752 1753 1754 1755 1756 1757 1758 1759

863

829

79 83 151 277 1663 13 37 17 1667 139 1669 167 557 19 239 31 67 419 43 839 73 7 82 58 17 421 337 281 241 211 563 26 89 47 1693 22 113 53 1697 283 1699 17 12 37 131 71 31 853 569

61 1709 19 59 107 571 857 21 13 101 859 191 43 1721 41 1723 431 23

1760

1761 1762 1763 1764 1765 1766 1767 1768 1769 1770 1771 1m 1m 1774 1775 1776 1777 1778 1779 1780 1781 1782 1783 1784 1785 1786 1787 1788 1789 1790 1791 1792 1793 1794 1795 1796 1797 1798 1799 1800

157 9 19 173 577 433 1733 34 347 31 193 79 47 29 1741 67 83 109 349 97 1747 23 53 15 103 73 1753 877 13 439 251 293 1759 11 587 881 43 14 353 883

31 17 61 59 23 443 197 887 71 37 1777 127 593 89 137 11 1783 223 17 47 1787 149 1789 179 199 10 163 23 359 449 599 31 257 10

The s..randaehe Function 5(n). n

5(n)

S(n)

5(n)

S(n)

1801 1802 1803 1804 1805 1806 1807 1808 1809 1810 1811 1812 1813 1814 1815 1816 1817 1818 1819 1820 1821 1822 1823 1824 1825 1826 1827 1828 1829 1830 1831 1832 1833 1834 1835 1836 1837 1838 1839 1840 1841 1842 1843 1844 1845 1846 1847 1848 1849 1850 1851 1852 1853 1854 1855 1856 1857 1858 1859 1860 1861 1862 1863 1864 1865 1866 1867 1868 1869 1870 1871 1872 1873 1874 1875

1801 53 601 41 38 43 139 113 67 181 1811 151 37 907 22 227 79 101 107 13 607 911 1823 19 73 83 29 457 59 61 1831

1876 1877 1878 1879 1880 1881 1882 1883 1884 1885 1886 1887 1888 1889 1890 1891 1892 1893 1894 1895 1896 1897 1898 1899 1900 1901 1902 1903 1904 1905 1906 1907 1908 1909 1910 1911 1912 1913 1914 1915 1916 1917 1918 1919 1920 1921 1922 1923 1924 1925 1926 1927 1928 1929 1930 1931 1932 1933 1934 1935 1936 1937 1938 1939 1940 1941 1942 1943 1944 1945 1946 1947 1948 1949 1950

67 1877 313 1879 47 19 941 269 157 29 41 37 59 1889 9 61 43 631 947 379 79 271 73 211 19 1901 317 173 17 127 953 1907 53 83 191 14 239 1913 29

1951 1952 1953 1954 1955 1956 1957 1958 1959 1960 1961 1962 1963 1964 1965 1966 1967 1968 1969 1970 1971 1972 1973 1974 1975 1976 1977 1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991

1951 61 31 977

2026 2017 2028 2029 2030 2031

409

2046

31 89 14

191 1051 701 263 421 13 43 31 37 211 2111 11 2113 151 47 46 73 353 163 53 101 1061 193 59 17 1063

2176 2177 2178 2179 2180 2181 2182 2183 2184 2185 2186 2187 2188 2189 2190 2191 2192 2193 2194 2195 2196 2197 2198 2199

17 311 22 2179 109 727 1091 59 13 23 1093 18 547 199 73 313 137 43 1097 439 61 39 157 733 11 71 367 2203 29 14 1103 2207 23 94 17 67 79 2213 41 443 277 739 1109 317 37 2221 101 19 139 89 53 131 557 743 223 97 31 29 1117 149 43 2237 373 2239 8 83 59 2243 17 449 1123 107 281 173 15

2251 2252 2253 2254 2255 2256 2257 2258 2259 2260 2261 2262 2263 2264 2265 2266 2267 2268 2269 2270 2271 2272 2273 2274 2275 2276 2277 2278 2279 2280 2281 2282 2283 2284 2285 2286 2287 2288 2289 2290 2291

2251 563 751 23 41 47 61 1129 251 113 19 29 73 283 151 103 2267 9 2269 227 757 71 2273 379 13

1163 179 97 137 233 37 53 2333

2045

2101 2102 2103 2104 210S 2106 2107 2108 2109 2110 2111 2112 2113 2114 2115 2116 2117 2118 2119 2120 2121 2122 2123 2124 2125 2126 2127 2128 2129 2130 2131 2132 2133 2134 2135 2136 2137 2138 2139 2140 2141 2142 2143 2144 2145 2146 2147 2148 2149 2150 2151 2152 2153 2154 2155 2156 2157 2158 2159

2326 2327 2328 2329 2330 2331 2332 2333 2334 2335 2336 2337 2338 2339 2340 2341 2342 2343

2044

1013 2027 26 2029 29 677 127 107 113 37 509 97 1019 2039 17 157 1021 227 73

229

47 131

367 17 167 919 613 23 263 307 97 461 41 71 1847 11 86 37 617 463

109 103 53 29 619 929 26

31 1861 19 23 233 373 311 1867 467 89 17 1871 13 1873 937 20

Page 4 of 8.

383

479 71 137 101 8 113 62 641 37 11 107 47 241 643 193 1931 23 1933 967 43 22 149 19 277 97 647 971 67 12 389

139 59 487 1949 13

1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004

2005 2006

2007 2008 2009

2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 2020 2021 2022 2023

2024 2025

163 103 89 653 14 53 109 151 491 131 983

281 41 179 197 73 29 1973 47 79 19 659 43 1979 11 283 991 661 31 397 331 1987 71 17 199 181 83 1993 997 19 499 1997 37 1999 15 29 13 2003 167 401 59 223 251 41 67 2011

2032

2033 2034 2035 2036 2037 2038 2039 2040 2041 2042 2043

2047 2048 2049 2050 2051 2052 2053

683

41 293

2060

19 2053 79 137 257 22 21 71 103

2061

229

2062

1031 2063 43 59 1033 53 47 2069 23 109 37 691 61 83 173 67 1039 11 13 2081 347

2054

2055 2056 2057 2058 2059

2063 2064

2065 2066 2067 2068

2069 2070 2071 2072 2073 2074 2075 2076 2077 2078 2079 2080 2081 2082 2083

2084 2085 2086

503

2087

61 53 31 8 2017 1009 673 101 47 337 34 23 10

2088

2089 2090

2091 2092 2093

2094 2095 2096 2097 2098 2099

2100

2083

521 139 149 2087 29 2089 19 41 523 23 349 419 131 233 1049 2099 10

2160

2161 2162 2163 2164 2165 2166 2167 2168 2169 2170 2171 2172 2173 2174 2175

709

19 2129 71 2131 41 79 97 61 89 2137 1069 31 107 2141 17 2143 67 13 37 113 179 307 43 239 269 2153 359 431 14 719 83 127 9 2161 47 103 541 433 38 197 271 241 31 167 181 53 1087 29

2200

2201 2202 2203 2204 2205

2206 2207 2208 2209 2210 2211 2212 2213 2214 2215 2216 2217 2218 2219 2220 2221 2222 2223 2224 2225 2226 2227 2228 2229 2230 2231 2232 2233 2234

2235 2236

2237 2238 2239 2240 2241 2242 2243 2244 2245 2246 2247 2248 2249 2250

2292

2293 2294 2295

2296 2297 2298 2299 2300 2301 2302 2303 2304 230S 2306 2307 2308 2309 2310 2311 2312 2313 2314 2315 2316 2317 2318 2319 2320 2321 2322 2323 2324 2325

569

23 67 53 19 2281 163 761 571 457 127 2287 13 109 229 79 191 2293 37 17 41 2297 383

22 23 59 1151 47 10 461 1153 769 577 2309

11 2311 34 257 89 463

193 331 61

29 211 43 101 83 31

389

2366

467 73 41 167 2339 13 2341 1171 71 293 67 23 2347 587 29 47 2351 14 181 107 157 31 2357 131 337 59 787 1181 139 197 43 26

2367

263

2368

37 103 79 2371 593 113 1187 19 11 2377 41 61 17 2381 397

2344

2345 2346 2347 234a

2349 2350 2351 2352 2353 2354 2355 2356 2357 2358 2359 2360

2361 2362 2363

2364

2365

2369 2370 2371 2372 2373 2374 2375 2376 2377 2378 2379 23aO

2381 2382

2383 2384 2385

2383

2388

149 53 1193 31 199

2389

2390 2391 2392 2393 2394 2395 2396 2397 2398 2399 2400

239 797 23 2393 19 479 599 47 109 2399 10

2386

2387

The s.erendache Function Sen).

Sen)

2401 2402 2403 2404 2405 2406 2407 2408 2409 2410 2411 2412 2413 2414 2415 2416 2417 2418 2419 2420 2421 2422 2423 2424 2425 2426 2427 2428

28 1201 89 601 37 401 83 43 73 241 2411 67 127 71 23 151 2417 31 59 22 269 173 2423 101 97 1213

2476 2477 2478 2479 2480 2481 2482 2483 2484 2485 2486 2487 2488 2489 2490 2491 2492 2493 2494 2495 2496 2497 2498 2499

619 2477 59 67 31 827 73 191 23 71 113 !29 311 131 83 53 89 277 43 499 13 227 1249 17 20 61 139 2503 313 167 179 109 19 193 251 31 157 359 419 503 37 839 1259

2551 2552 2553 2554 2555 2556 2557 2558 2559

2551 29 37 1277 73 71 2557 1279 853 12 197 61 233 641 19 1283 151 107 367 257 857 643 83 13 103 23 859 1289 2579 43 89 1291 41 19 47 431 199 647

2626 2627 2628 2629 2630 2631 2632 2633 2634 2635 2636 2637

101 71 73 239 263 877 47 2633 439 31 659 293 1319 29 11 139 1321 881 661 46 14 2647 331

2701 2702 2703 2704 2705 2706 2707 2708 2709 2710 2711 2712 2713 2714 2715 2716 2717 2718 2719 2720 2nl 2722 2723 2n4 2n5 2m 2n7 2m 2729 2730 2731 2732 2733 2734 2735 2736 2737 2738 2739 2740 2741 2742 2743 2744 2745 2746 2747 274.'5 2749 2750 2751 2752 2753 2754 2755 2756 2757 2758 2759 2760 2761 2762 2763 2764 2765 2766 2767 2768 2769 2770 2771 2m 2m 2774 2775

73 193 53 26 541 41 2707 677 43 271 2711 113 2713 59 181 97 19 151 2719 17 907 1361

2776 2m 2m 2779 2780 2781 2782 2783 2784 2785 2786 2787 2788 2789 27'90 2791 2792 2793 2794 2795 2796 2797 2798 2799 2800 2801 2802

347 2m 463 397 139 103 107 23 29 557 199

2851 2852 2853 2854 2855 2856 2857 2858 2859

2851 31 317 1427 571 17 2857 1429 953 13 2861 53 409 179 191 1433 61 239 151 41 29 359 26 479 23 719 137 1439 2879 8 67 131 62 103 577 37 2887 38 107 34 59 241 263 1447 193 181 2897 23 223 29 967 1451 2903 22 83 1453 19 n7

"2429

2430 2431 2432 2433 2434 2435 2436 2437 2438 2439 2440 2441 2442 2443 2444 2445 2446 2447 2448

2449 2450 2451 2452 2453 2454 2455 2456 2457 2458 2459 2460 2461 2462 2463 2464 2465 2466 2467 2468 2469 2470 2471 24n 2473 2474 2475 Page 5

809

607 347 12 17 19 811 1217 4.'57 29 2437 53 271 61 2441 37 349 47 163 1223 2447 17 79 14 43 613 223 409 491 307 13 1229 2459 41 107 1231 821 11 29 137 2467 617 823

19 353 103 2473 1237 11

ot 8.

2500

2501 2502 2503 2504 2505 2506 2507 2508 2509 2510 2511 2512 2513 2514 2515 2516 2517 2518 2519 2520 2521 2522 2523 2524 2525 2526 2527 2528 2529 2530 2531 2532 2533 2534 2535 2536 2537 2538 2539 2540

2541 2542

2543 2544 2545 2546

2547 2548 2549 2550

229

7 2521 97 58 631 101 421 38 79 281 23 2531 211 149 181 26 317 59 47 2539 127 22 41 2543 53 509 67 283 14 2549 17

2560

2561 2562

2563

2564 2565 2566

2567 2568 2569

2570 2571 25n 2573 2574 2575 2576 2577 2578 2579 2580 2581 258Z

2583 2584 2585 2586

2587 2588 2589 2590 2591 2592 2593 2594 2595 2596 2597 2598 2599 2600 2601 2602 2603 2604 2605

2606 2607 2608 2609 2610 2611 2612 2613 2614 2615 2616 2617 2618 2619 2620 2621 2622 2623 2624 2625

863

37 2591 9 2593 1297 173 59 53 433 113 13 34 1301 137 31 521 1303 79 163 2609 29 373 653 67 1307 523 109 2617 17 97 131 2621 23 61 41 15

2638

2639 2640

2641 2642 2643 2644 2645 2646 2647 2648 2649 2650 2651 2652 2653 2654 2655 2656 2657 2658

2659 2660 2661 2662 2663 2664 2665 2666

2667 2668 2669

2670 2671 26n 2673 2674 2675 2676 2677 2678 2679 2680 2681 2682 2683 2684

2685 2686

2687 2688 2689 2690

2691 2692 2693 2694 2695 2696 2697

883 53

241 17 379 1327 59 83 2657 443 2659 19 887 33 2663 37 41 43 127 29 157 89 2671 167 12 191 107 223

2677 103 47 67 383

149 2683 61 179 79 2687 8 2689

269 23 673 2693 449 14 337 31 71

2698 2699

2699

2700

389

227 109 47 101 31 2729 13 2731 683 911 1367 547 19 23 74 83 137 2741 457 211 21 61 1373 67 229 2749 15 131 43 2753 17 29 53 919 197 89 23 251 1381 307 691 79 461 2767 173 71 277 163 11 59 73 37

2803

2804 2805 2806

2807 2808

2809 2810 2811 2812 2813 2814 2815 2816 2817 2818 2819 2820 2821 2822 2823 2824 2825

2826 2827 2828 2829 2830

2831 2832 2833 2834 2835 2836

2837 2838 2839 2840 2841 2842 2843 2844

2845 2846 2847 2848 2849 2850

929

41 2789 31 2791 349 19 127 43 233 2797 1399 311 10 2801 467 2803 701 17 61 401 13 106 281 937 37 97 67 563

11 313 1409 2819 47 31 83 941 353 113 157 257 101 41 283 149 59 2833 109 9 709 2837 43 167 71 947 29 2843 79 569 1423 73

89 37 19

2860

2861 2862 2863 2864 2865 2866 2867 2868 2869

2870 2871 28n 2873 2874 2875 2876 2877 2878 2879 2880

2881 288Z 2883

2884 2885 2886

2887 2888 2889

2890 2891 2892

2893 2894 2895 2896

2897 2898

2899 2900

2901 2902 2903 2904

2905 2906 2907 2908 2909

2909

2910 2911 2912 2913 2914 2915 2916 2917 2918 2919 2920 2921 2922 2923 2924 2925

97 71 13 971 47 53 15 2917 1459 139 73 127 4.'57 79 43 13

n 2926

2927 292!

2929 2930 2931 2932 2933 2934 2935 2936

2937 2938

2939 2940

2941 2942 2943 2944 2945 2946

2947 2948

2949 2950 2951 2952 2953 2954 2955 2956

2957 2958 2959 2960

2961 2962 2963 2964

2965 2966

2967 2968

2969 2970 2971 29n 2973 2974 2975 2976 2977 2978 2979 2980

2981 2982 2983 2984

2985 2986

2987 2988

2989 2990

2991 2992 2993 2994 2995 2996

2997

Sen) 19 2927 61 101 293 977 733 419 163 587 367 89 113 2939 14 173 1471 109 23 31 491 421 67 983

59 227 41 2953

211 197 739 2957 29 269 37 47 14.'51 2963

19 593 1483 43 53 2969 11 2971 743 991 1487 17 31 229

1489 331 149 271 71 157 373 199 1493 103 83 61 23 997 17 73 499 599 107 37 1499

2998 2999

2999

3000

The s.erandeche Function Sen). n

Sen)

3001 3002

3001 79 13 751 601 167 97 47 59 43 3011 251 131 137 67 29 431 503 3019 151 53 1511

3076 30n 3078 3079 3080 3081

3003 3004

3005 3006

3007 3008 3009

3010 3011 3012 3013 3014 301i 3016 3017 3018 3019 3020 3021 3022 3023

3024 3025

3026 3027

.3028 3029 3030

3031 3032 3033 3034 3035 3036

3037 3038

3039 3040

3041 3042 3043 3044

3045 3046 3047 3048

3049 3050 3051 3052 3053 3054

3055 3056

3057 3058

3059 3060

3061 3062

3063 3064 3065

3066

3067 3068

3069 3070 3071 30n 3073 3074 3075

3023

9 22 89 1009 757 233 101 433 379 337 41 607 23 3037 31 1013 19 3041 26 17'9 761 29 1523 2n 127 3049 61 113 109 71 509 47 191 1019 139 23 17 3061 1531 1021 383

613 73 3067 59 31 307 83 12 439 53 41

Page 6 of 8.

Sen) 7~

3082

181 19 3079 11 79 67

3083

3084

257 617 1543 21 193 3089 103 281

3085 3086 3087 3088 3089 3090

3091 3092 3093 3094 3095

3096

3097 3098 3099

3100 3101 3102 3103 3104 3105 3106 3107 3108 3109 3110 3111 3112 3113 3114 3115 3116 3117 3118 3119 3120 3121 3122 3123 3124 3125 3126 3127 3128 3129 3130 3131 3132 3133 3134 3135 3136 3137 3138 3139 3140 3141 3142 3143 3144 3145 3146 3147 3148 3149 3150

773

1031 17 619 43 163 1549 1033 31 443 47 107 97 23 1553 239 37 3109 311 61 389

283 173 89 41 1039 1559 3119 13 3121 223 347 71 25 521 59 23 149 313 101 29 241 1567 19 14 3137 523 73 157 349 1571 449 131 37 22 1049 787 67 10

Sen)

3151 3152 3153 3154 3155 3156 3157 3158 3159 3160 3161 3162 3163 3164 3165 3166 3167 3168

137 197 1OS1 83 631 263 41 1579 13 79 109 31 3163 113 211 1583 3167 11

31~

3170 3171 3172 3173 3174 3175 3176 31n 3178 317'9 3180 3181 3182 3183 3184 3185 3186 3187 3188 3189 3190 3191 3192 3193 3194 3195 3196 3197 3198 3199 3200 3201 3202 3203 3204 3205 3206 3207 3208 3209 3210 3211 3212 3213 3214 3215 3216 3217 3218 3219 3220 3221 3222 3223 3224 3225

317 151 61 167 46 127 397 353 227 34 53 31&1 43 1061 199 14 59 3187 7'97 1063 29 3191 19 103 1597 71 47 139 41 457 10 97 1601 3203 89 641 229 1~

401 3209 107 26 73 17 1607 643 67 3217 1609 37 23 3221 17'9 293 31 43

3226 3227 3228 3229 3230 3231 3232 3233 3234 3235 3236 3237 3238

3239 3240 3241 3242 3243 3244 3245 3246 3247 3248 3249 3250 3251 3252 3253 3254 3255 3256 3257 3258

3259 3260 3261 3262 3263 3264 3265 3266 3267 3268 3269 3270 3271 32n 3273 3274 3275 3276 32n 3278 327'9 3280 3281 3282 3283 3284

3285 3286 3287 3288

Sen)

1613 461 269 3229 19 359 101 61 14 647

3301 3302 3303

3301 127 367 59 661 29 3307 827 1103 331 43 23 3313 1657 17

3376 33n 3378 3379 3380 3381

3451 3452 3453 3454 3455 3456 3457 3458 3459

809

83 1619 7'9 9 463

1621 47 811 59 541 191 29 38

15 3251 271 3253 1627 31 37 3257 181 3259 163 1087

3293

89 61 659 103 157 97 3299 11

3300

3310 3311 3312 3313 3314 3315 3316 3317 3318 3319 3320 3321

823

829

3324 3325 3326 3327 3328

107 79 3319 83 41 151 3323 2n 19 1663 1109 13

3329

3330

3322 3323

3344

37 3331 17 101 1667 29 139 71 1669 53 167 257 557 3343 19

3345

223

3346

239 3347 31 197 67 1117 419 479 43 61 839 373 73 3359 8 3361 82 59 58 673 17 37 421 1123 337 3371 281 3373 241 15

3331 ID2 3333 3334

3335 3336

ID7

3299

3307 3308 3309

3338

47 1097

3296 3297 3298

3306

251 17 653 71 22 43 467 109 3271 409 1091 1637 131 13 113 149 1093 41 193 547 67 821 73 53 173 137

3289

3295

3305

233

3290 3291 3292 3294

3304

ID9 3340

3341 3342

3343

3347 3348

3349 3350 3351 3352 3353 3354 3355 3356 3357 3358 3359 3360

3361 3362 3363 3364 3365 3366

3367 3368 ~

3370 3371 33n 3373 3374 3375

3388

211 307 563 109 26 23 89 199 47 6n 1693 1129 22

3389

3390 3391 3392 3393 3394 3395 3396 3397 3398 3399

113 3391 53 29

3382 3383 3384

3385 3386

3387

3400

3401 3402

3403 3404 3405

3406

3407 3408 3409

3410 3411 3412 3413 3414 3415 3416 3417 3418 3419 3420 3421 3422 3423 3424 3425 3426 3427 3428 3429 3430 3431 3432 3433 3434 3435 3436 3437 3438 3439 3440 3441 3442 3443

3463

1~7

97 283 7'9 1699 103 17 179 12 83 37 227 131 3407 71 487 31 379 853 3413 569

3470 3471 34n 3473 3474 3475 3476 34n 3478 3479

683

3490 3491 3492 3493 3494 3495 3496 3497 3498 3499 3500 3501 3502 3503 3504 350S 3506 3507 3508 3509 3510 3511 3512 3513 3514 3515 3516 3517 3518 3519 3520 3521 3522 3523 3524 3525

347 89 31 151 193 139 7'9 61 47 71 29 118 1741 43 67 41 83 317 109 1163 349 3491 97 499 1747 233 23 269 53 3499 15

3446

383

3448

431 3449 23

3449 3450

3463 3464

1151 157 691 9 3457 19 1153 173 3461 5n

3468

3447

3444

3461 3462

29 863

433 11 1733 3467 34

61 67 1709 263 19 311 59 163 107 137 571 149 857 127 21 73 13 3433 101 229 859 491 191 181 43 37 lnl 313 41 53 1m

3445

3460

Sen)

3465 3466

3467

3480

3481 3482 3483 3484 3485 3486

3487 3488 3489

389

103 113 73 701 1753 167 8n 29

13 3511 439 1171 251 37 293 3517 1759 23 11 503 587 271 881 47

Sen)

3526 3527 3528 3529 3530 3531 3532 3533 3534 3535 3536 3537 3538 3539 3540 3541 3542 3543 3544 3545 3546 3547 3548 3549 3550 3551 3552 3553 3554 3555 3556 3557 3558 3559 3560 3561 3562 3563 3564 3565 3566 3567 3568

43 3527 14 3529 353 107

35~

3570 3571 35n 3573 3574 3575 3576 35n 3578 3579 3580 3581 3582 3583 3584 3585 3586 3587 3588 3589 3590 3591 3592 3593 3594 3595 3596 3597 3598 3599 3600

883

3533 31 101 17 131 61 3539 59 3541 23 1181 443 709

197 3547 887 26 71 67 37 19 1m 7'9 127 3557 593 3559 89 1187 137 509 11 31 1783 41 223 83 17 3571 47 397 1787 13 149 73 1789 1193 179 3581 199 3583 12 239 163 211 23 97 359 19 449 3593 599 719 31 109 257 61 10

The Sllerandac:he FlrCtion

Sen).

Sen)

3601 3602

277 1801 1201 53 103 601 3607 41 401 38 157 43 3613 139 241 113 3617 67 47 181 71 1811

3676 3677 3678 3679

3603 3604 3605

3606

3607 3608 3609

3610 3611 3612 3613 3614 3615 3616 3617 3618 3619 3620

3621 3622 3623

3624

3623 lSI

3633 3634

29 37 31 907 191 22 3631 227 173 79

3635

3636

101 3637 107 1213 13 331 607 3643 911 15 1823 521 19

3625 3626

3627 3628 '3629 3630

3631 3632

3637 3638

3639 3640

3641 3642

3643 3644 3645 3646

3647 3648

3649 3650 3651 3652 3653 3654 3655 3656 3657 3658 3659 3660

3661 3662 3663

3664 3665 3666

3667 3668 3669

3670 3671 36n

3673 3674 361S p~

73 1217 83 281 29 43 457 53 59 3659 61 523 1831 37 229 733

47 193 131 1223 367 3671 17 3673 167 14

7 of 8.

, n

3680

919 3677 613 283 23

3681

409

3756

3682

31S7

3685

263 127 307 67

3686

3687

1229 461 31 41 3691 71 1231 1847 739 11 3697 86 137 37 3701 617 46

3683 3684

3688 3689 3690

3691 3692 3693 3694 3695 3696

3697 3698

3699 3700 3701 3702 3703 3704 3705 3706 3707 3708 3709 3710 3711 3712 3713 3714 3715 3716 3717 3718 3719 3720 3nl 3n2

3723 3n4 3n5 3n6 3n7 3n8

3729 3730 3731 3732 3733 3734 3735 3736 3737 3738 3739 3740 3741 3742 3743 3744 3745 3746 3747 3748 3749 31S0

463

19 109 337 103 3709 53 1237 29 79 619 743 929 59 26

3719 31 122 1861 73 19 149 23 3n7 233

113 373 41

311 3733 1867 83 467 101 89

3739 17 43 1871 197 13 107 1873 1249 937 163 20

31S1 31S2

3m 3154 31S5

3758 31S9

3760 3761 3762 3763 3764 3765 3766 3767 3768 3769 3770 3m 3m 3m

3774 377S 3776 3m 3m 3779 3780 3781 3782 3783 3784 3785 3786

3787 3788 3789 37'90 3791 3792 3793 37'94 3795 3796 3797 37'98 3799 3800

3801 3802 3803 3804 3805 3806

3807 3808 3809

3810 3811 3812 3813 3814 3815 3816 3817 3818 3819 3820

3821 3822 3823

3824 3825

Sen)

31 67 139 1877 1S1 313 34 1879 179 47 3761 19 71 941 251

3826 3827

1913

3828 3829 3830

29 547 383 1277 479

3901 3902 3903 3904 3905 3906 3907 3908 3909 3910 3911 3912 3913 3914 3915 3916 3917 3918 3919 3920 3921 3922 3923 3924 3925 3926 3927 3928 3929 3930 3931 3932 3933 3934 3935 3936 3937 3938 3939 3940 3941 3942 3943 3944 3945 3946 3947 3948 3949 3950 3951 3952 3953 3954 3955 3956 3957 3958 3959 3960 3961 3962 3963 3964 3965 3966 3967 3968 3969 3970 3971

83 1951 1301 61 71 31 3907 977 1303 23 3911 163 43 103 29

3976 3977 3978 3979 3980 3981 3982 3983 3984 3985 3986 3987 3988 3989 3990 3991 3992 3993 3994 3995 3996 3997 3998 3999

71 97 17 173 199 1327 181

4051 4052 4053 4054 4055 4056 4057 4058 4059

4051 1013 193 2027 811 26 4057 2029 41 29 131 677 239 127 271 107 83 113 313 37 59 509 4073 97 163 1019 151 2039 4079 17 53 157 1361 1021 43 227 67 73 47 409 4091 31

4126 4127 4128 4129 4130 4131 4132 4133 4134 4135 4136 4137 4138 4139 4140 4141 4142 4143 4144 4145 4146 4147 4148 4149 4150 4151 4152 4153 4154 4155 4156 4157 4158 4159 4160 4161 4162 4163 4164 4165 4166 4167 4168 4169 4170 4171

2063 4127 43 4129 59 17 1033 4133 53 827 47 197

269

3767 157 3769 29 419 41 21 37 lSI

59 1259 1889 3779 9 199 61 97 43 1S7 631 541 947 421 379 223

79 3793 271 23 73 3797 211 131 19 181 1901 3803 317 761 173 47 17

3831 3832 3833 3834

3835 3836

3837 3838

3839 3840

3841 3842 3843 3844

3833

59 137 1279 101 349 10 167 113 61 62

3845

769

3846

3860

641 3847 37 1283 11 3851 107 3853 47 257 241 29 643 227 193

3861

3862 3863

1931 3863

3864

3847 3848

3849 3850 3851 3852 3853 3854

3855 3856

3857 3858 3859

3865

3866

3888

1933 1289 967 73 43 79 22 1291 149 31 19 3877 277 431 97 3881 647 353 971 37 67 26 12

3889

3890

3867 3868

3869 3870 3871 38n

3873 3874 381S 3876 3877 3878 3879 3880

3881 3882

3883

293

3884

127 103 953 41 1907 109 53 347 83 67 191 3821 14

3885

3823

3898 3899

239 17

3886

3887

3892

389 1297 139

3893

229

3894 3895 3896

3891

3897 3900

487 433 1949 557 13

39n

3973 3974 391S

3917 653 3919 14 1307 53 3923 109 157 151 17 491 3929 131 3931 983

23 281 787 41 127 179 101 197 563

73 3943 29 263

1973 3947 47 359 79 439 19 67 659 113 43 1319 1979 11>7 II 233

283 1321 991 61 661 3967 31 14 397 38 331 137 1987 53

569

4001 4002

83 797 1993 443 997 3989 19 307 499 33 1997 47 37 571 1999 43 15 4001 29

4003

4060

4061 4062 4063 4064

4065 4066

4067 4068 4069 4070 4071 40n

4004

4073 4074 401S 4076 4077 4078 4079

4005

89 2003

4081

4000

4006 4007 4008 4009

4010 4011 4012 4013 4014 4015 4016 4017 4018 4019 4020 4021 4022 4023 4024 4025 4026 4027 4028 4029 4030 4031 4032 4033 4034

4035 4036 4037

4007 167 211 401 191 59 4013 223 73 251 103 41 4019 67 4021 2011 149 503 23 61 4027 53 79 31 139 8 109 2017 269

4044

1009 367 673 577 101 449 47 311 337

4045

809

4046

4047 4048 4049 4050

4038

4039 4040 4041 4042 4043

23 4049 10

4080

4082

4083 4084

4085 4086

4OS7 4088

4089 4090

4091 4092

4093 4094

4093

4095 4096

13 16 241

4097 4098 4099

4100 4101 4102 4103 4104 4105 4106 4107 410S 4109 4110 4111 4112 4113 4114 4115 4116 4117 4118 4119 4120 4121 4122 4123 4124 4125

683

4099 41 1367 293

373 19 821 2053 74 79 587 137 4111 257 457 22 823 21 179 71

1373 103 317 229 31 1031 15

41n

4173 4174 411S

4176 4177 4178 4179 4180 4181 4182 4183 4184 4185 4186 4187 4188 4189 4190 4191 4192 4193 4194 4195 4196 4197 4198 4199 4200

2069

4139 23 101 109 1381 37 829

691 29 61 461 83 593 173 4153 67 277 1039 4157 II

4159 13 73 2081 181 347 17 2083 463

521 379 139 97 149 107 2087 167 29 4177 2089 199 19 113 41 89

523 31 23 79 349 71 419 127 131 599 233 839 1049 1399 2099 19 10

The s.arandache Function 5(n). n

4201 4202 4203

4204 4205 4206

4207 4208 4209

4210 4211 4212 4213 4214 4215 4216 4217 4218 4219 4220

4221 4222 42Z3

4224 42Z5 4226 4227 4228 42Z9 4230

4231 4232 4233

4234

4235 4236

4237 4238 4239 4240 4241 4242 4243 4244 4245 4246 4247 4248 4249 4250 4251 4252 4253 4254 4255 4256 4257 4258

4259 4260

4261 4262 4263

4264 4265 4266 4267 4268 4269 4270 4271 42n 4273 4274 4275

5(n)

$(n)

5(n)

4201 191 467 1051 58 701 601 263 61 421 4211 13 383 43 281 31 4217 37 4219 211 67 2111 103 11 26 2113 1409

4276 4277 4278 4279 4280 4281 4282 4283

1069 47 31 389 107 1427 2141 4283 17 857 2143 1429 67 4289 13 613 37 53 113 859 179 4297 307 1433 43 23 239 331 269 41 2153 73 359 139 431 479 14 227 719

4351 4352 4353 4354 4355 4356 4357 4358 4359 4360 4361 4362 4363 4364 4365 4366 4367 4368 4369 4370 4371 43n 4373 4374 4375 4376 4377 4378 4379 4380 4381 4382 4383 4384 4385 4386 4387 4388 4389 4390 4391 4392 4393 4394 4395 4396 4397 4398 4399 4400 4401

229 17 1451 311 67 22 4357 2179 1453 109 89 n7 4363 1091 97 59 397 13 257 23 47 1093 4373 18 20 547 1459 199 151 73 337 313 487 137 877 43 107 1097 19 439 4391 61 191 39

4426 4427 4428 44Z9 4430 4431 4432 4433 4434 4435 4436 4437

2213 233 41 103 443 211 277 31 739 887 1109 29 317 193 37 4441 2221 1481 101 127 19 4447 139 1483 89 4451 53 73 131 11 557 4457 743 21 223 1487

643 2251 79 563 53 751 4507 23 167 41 347 47 4513 61 43 1129 4517 251 4519 113 137 19 4523 29 181 73 503

4576 4577 4578 4579 4580 4581 4582 4583 4584 4585 4586 4587 4588 4589 4590 4591 4592 4593 4594 4595 4596 4597 4598 4599

13 199 109 241 229 509 79 4583 191 131

4651 4652 4653 4654 4655 4656 4657 4658 4659

ZZ93

4661 466Z

4651 1163 47 179 19 97 4657 137 1553 233 79 37

4663

4664

4667

53 311 Z333 359

4668

389

283

4603 4604 4605

4669 4670 4671 46n 4673 4674 4675 4676 4677 4678 4679

29 467 173 73 4673 41 17 167 1559 2339 4679 13 151 2341 223 1171 937 71 109

4470 4471 44n 4473 4474 4475 4476 4477 4478 4479

4n6 4n7 4n8 4729 4730 4731 4732 4733 4734 4735 4736 4737 4738 4739 4740 4741 4742 4743 4744 4745 4746 4747 4748 4749 4750 4751 4752 4753 4754 4755 4756 4757 4758 4759 4760 4761 4762 4763 4764 4765 4766 4767 4768 4769 4770 4771 4m 4m 4774 4775 4776 4777 4778 4779 4780 4781 4782 4783 4784 4785 4786 4787 4788 4789 4790 4791 4792 4793 4794 4795 4796 4797 4798 4799

139 163 197 4729 43 83 26 4733 263 947 37 1579 103 677 79 431 2371 31 593 73 113 101 1187 1583 19 4751 11 97 2377 317 41 71 61 4759 17 46 2381 433 397 953

293

4501 4502 4503 4504 4505 4506 4507 4508 4509 4510 4511 4512 4513 4514 4515 4516 4517 4518 4519 4520 4521 4522 4523 4524 4525 4526 4527 4528 4529 4530 4531 4532 4533 4534 4535 4536 4537 4538 4539 4540 4541 4542 4543 4544 4545 4546 4547 4548 4549 4550 4551 4552 4553 4554 4555 4556 4557 4558 4559 4560 4561 4562 4563 4564 4565 4566 4567 4568 4569 4570 4571 45n 4573 4574 4575

lSI 42Z9

47 4231 46 83 73 22 353 223 163 157 53 4241 101 4243 1061 283 193 137 59 607 17 109 1063 4253 709

37 19 43 2129 4259 71 4261 2131 29 41 853 79 251 97 1423 61 4271 89 4273 2137 19

Page 8 of 8.

4284

4285 4286 4287 4288 4289 4290 4291 4292 4293 4294 4295 4296 4297 4298 4299

4300 4301 4302 4303 4304 4305

4306 4307 4308 4309 4310 4311 4312 4313 4314 4315 4316 4317 4318 4319 4320 4321 4322 4323 4324 4325 4326 4327 4328 4329 4330 4331 4332 4ID 4334 4335 4336 4337 4338 4339 4340 4341 4342 4343 4344 4345 4346 4347 4343 4349 4350

863

83 1439 127 617 9 149 2161 131 47 173 103 4327 541 37 433 71 38 619 197 34 271 4337 241 4339 31 1447 167 101 181 79 53 23 1087 4349 29

4402 4403

4404 4405

4406 4407 4408 4409

4410 4411 4412 4413 4414 4415 4416 4417 4418 4419 4420 4421 44Z2 4423 4424 44Z5

157 4397 733 83 11 163 71

37 367 881 2203 113 29 4409 14 401 1103 1471 2207 883

23 631 94 491 17 4421 67 4423 79 59

4438

4439 4440

4441 4442 4443 4444

4445 4446

4447 4448

4449 4450 4451 4452 4453 4454 4455 4456 4457 4458 4459 4460

4461 4462 4463 4464

4465 4466

4467 4468 4469

4480

4481 4482 4483 4484 4485 4486

4487 4488 4489

4490 4491 4492 4493 4494 4495 4496 4497 4498 4499 4500

97 4463

31 47 29 1489 1117 109 149 263 43 71 2237 179 373 37 2239 1493 8 4481 83 4483

59 23 2243 641 17 134 449 499 1123 4493 107 31 281 1499 173 409 15

647 151 197 103 1511 2267 907 9 349 2269 89

227 239 757 59 71 101 2273 4547 379 4549 13 41 569 157 23 911 67 31 53 97 19 4561 2281 26 163 83 761 4567 571 1523 457 653 127 269 2287 61

4600

4601 4602

4606

4607 4608 4609

4610 4611 4612 4613 4614 4615 4616 4617 4618 4619 4620

4621 4622 46Z3

4624 46Z5 4626

4627 4628 46Z9 4630

4631 4632

4633 4634

4635 4636

4637 4638

4639 4640

4641 4642 4643 4644

4645 4646

4647 4648

4649 4650

139 37 353 17 4591 41 1531 2297 919 383 4597 22 73 23 107 59 4603

1151 307 47 271 12 419 461 53 1153 659 769 71 577 19 2309

149 11 4621 2311 67 34 37 257 661 89 1543 463 421 193 113 331 103 61 4637 m 4639 29 17 211 4643 43 929 101 1549 83 4649 31

4660

4665 4666

4680

4681 4682

4683 4684

4685 4686

4687 4688 4689

4690 4691 4692

4693 4694 4695 4696 4697 4698 4699 4700 4701 4702 4703 4704 4705 4706 4707 4708 4709 4710 4711 4712 4713 4714 4715 4716 4717 4718 4719 4nO 4n1 4722

4723 4n4 4725

293

521 67 4691 23 38

2347 313 587 61 29 127 47 1567 2351 4703 14 941 181 523 107 277 157 673 31 1571 2357 41 131 89 337 22 59 4nl 787 4723 1181 10

4800

2383

227 149 251 53 367 1193 43 31 191 199 281 Z389

59 239 683

797 4783 23 29 2393 4787 19 4789 479 1597 599 4793 47 137 109 41 2399 4799 10

'SNP _TAB, H. Ibstedt, 930322 'This program tabulates S(n) for n =fY'J, 1 < P < 74, 1 <J, 76, using data form the file SNP_ASC. OEFLNG I-S :OIM KP(1575),KJ(1575),SP(1575) CLS: WIOTH "LPT1 :",120 OPEN "SNP.OAr FOR INPUT AS #1 FOR 1=1 TO 1575 INPUT #1 ,KP(I),KJ(I),SP(I) NEXT CLOSE #1 51S=lI j 54$="1 P(l> 51$="1 B1S="j 53$=" 56S=" P(l > 59$=" 83S="

:52$=" I ID(I,J>I" : 55$=" P( I) I In :58$=" I '11 :82$=" j

i ID(l,J)I" i

I" '11

ID(l,J>I"

III I II

11 =1 :12=75 :P1 =1 LW: LPRINT TA8(12) "The Smarandache Function S(n) = O(I,J) for powers of primes P(I)"J." LPRINT TA8(12) S1$; :FOR 1=1 TO 3 :LPRINT S2$; :NEXT :LPRINT S3$; LPRINT TA8(12) S4$; :FOR 1=1 TO 3 :LPRINT S5$; :NEXT :LPRINT S6$; LPRINT TA8(12) S7$; :FOR 1=1 TO 3 :LPRINT S8$; :NEXT :LPRINT S9$; FOR 1=11 TO 12 LPRINT TA8(12) "I"; FOR J=OTO 4 LPRINT USING "#####";KP(I +J*75); LPRINT" I"; :LPRINT USING "#####";KJ(I+J*75); :LPRINT" I"; :LPRINT USING "#####";SP(I +J*75); LPRINT" I" ; NEXT NEXT LPRINT TA8(12) 81$; :FOR 1=1 TO 3 :LPRINT 82$; :NEXT :LPRINT 83$; LPRINT TA8(12) 路Page"P1 "of 3. LPRINT CHR$(12) IF P1 = 1 THEN P1 =2 :11 =601 :12=675 :GOTO LW IF P1 =2THEN P1 =3 :11 =1201 :12=1275 :GOTO LW PRINT "ENO" :ENO

The Smerandache Function S(n)=O(I,J) for powers of primes P(I)"J. PO)

2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 3D 31 32 33

34 35 36 37 38 39 40 41 42 43

44 45 46

47 48

49 50 51 52 53 54 55 56 57 58

59 60 61 62 63

64 65 66 67 68

69 70 71

73 74 75

Page 1 of 3.

D( I,J)

2 4 4 6 8 8 8 10 12 12 14 16 16 16 16 18 20 20 22 24 24 24 26 28 28

3D 32 32 32 32 32 34 36 36 38

40 40 40 42 44 44 46 48 48 48

48 50 52 52 54 56 56 56 58 60 60 62 64 64 64 64 64 64 66 68 68 70

n n n

74 76 76 78

P(I>

3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3

O( I,J)

1 2 3 4 5 6 7

a 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33

34 35 36 37 38 39 40 41 42 43 44 4S

3 6 9 9 12 15 18 18 21 24 27 27 27 3D 33 36 36

39 42 45 45 48

51 54 54 54 57 60 63 63

66 69 72 72 75 78

81 81 81 81 84

87 90 90 93

99 99 102

49 50 51 52 53 54 55 56 57 58

59 60 61 62 63

64 6S 66 67 68 69

70 71

73 74 75

lOS 108 loa loa 111 114 117 117 120 123 126 126 129 132 135 135 135 138 141 144 144 147 150 153 153 156

P(I)

5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5

D(I,J)

1 2 3 4 5 6 7

5 10 15 20 25 25 3D 35 40 45 50 50 55 60 65 70 75 75 ao 85 90 95 100 100 lOS 110 115 120 125 125 125 13D 135 140 145 150 150 155 160 165 170 175 175 lao 185 190 195

200

49 50 51 52 53 54 55 56 57

200 20S 210 215

a 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28

29 3D 31 32 33

34 35 36 37 38 39 40 41 42 43

44 45 46

59 60 61 62 63

64 65 66 67 68 69

70 71 72 73 74 75

220

225 225 230 235 240 245 250 250 250 255 260 265 270 275 275 280

285 290 295 300 300 305

PO)

7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7

1 2 3 4 5 6 7

a 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 3D 31 32 33

34 35 36 37 38 39 40 41 42 43 44 45 46

47 48

49 50 51 52 53 54 55 56 57 58 59 60 61 62

O(l,J)

P(I>

7 14 21 28 35 42 49 49 56

11 11 11

70 77 84

91 98 98 lOS 112 119 126 133 140 147 147 154 161 168 175 182 189 196 196 203 210 217 224 231 238

245 245 252 259 266 273 2ao 287 294 294

3Dl 308 315 322 329 336 343 343 343 350 357 364

371 378

38S

64 65 66 67 68 69 70 71 72 73 74 75

392 392 399 406 413 420 427 434 441 441 448

455

"1111 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11

"1111 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11

"1111 11 11 11 11 11 11 11 11 11 11

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33

34 35 36 37 38 39 40 41 42 43 44 45 46

0(1, J)

11 22 33

44 55 66 77 88

99 110

121 121 132 143 154 165 176 187 198 209

220 231 242 242 253 264 275 286

297 3D!

319 33D 341 352 363 363

374 385 396 407 418 429 440 451 462

473

484

49 50 51 52 53 54 55 56 57 58 59 60 61 62

495 506 517 528 539 550 561 572 583

594 605 605

616 627

638

64 65 66 67 68

649

660

671 682

693 704

70 71

715 726 726

737

74 75

748 759

The s.arandache Function S(n)-D(I,Jl for powers of pri=es P(I)'J, P(ll

Z3 Z3 Z3 Z3 Z3 Z3 Z3 Z3 23 23 Z3 23 23 23 Z3 23 23 Z3 Z3 23 23 23 Z3 23 Z3 23 Z3 Z3 23 Z3 Z3 23 Z3 23 23 Z3 23 23 Z3 23 Z3 23 Z3 23 23 Z3 Z3 23 Z3 Z3 Z3 23 Z3 Z3 23 Z3 Z3 Z3 Z3 Z3 23 Z3 Z3 Z3 Z3 Z3 Z3 23 Z3 23 Z3 Z3 23 Z3 Z3 Page 2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 Z3 24 25 26 27 28

O(!,Jl

Z3 46 69 92 115 138 161 184 207 230

253 276 299 322 345 368

391 414 437 460 483

506 529 529 552 5~

598 621

29 30 31

644

32 33

713 736

34 35 36 37

39 40 41 42 43

44 45 46 47 48 49 50

51 52 53 54 55 56 57

sa 59 60 61 62 63 64 65 66 67 68 69

70 71 72 73 74 ~

of 3,

667 690

782 805

828 851 874 897 920

943 966 989

1012 1035 1058 1058 1081 1104 1127 1150 1173 1196 1219 1242 1265 1288 1311 1334 1357 1380 1403 1426 1449 1472 1495 1518 1541 1564 1587 1587 1610 1633 1656

P(ll

29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28

29 30 31 32 33

34 35 36 37 38

39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57

59 60 61 62 63 64 65 66 67 68 69

0(1, Jl 29

87 116 145 174 203 232 261 Z90 319 348

377 406 435 464

493 522 551 580 609 638

667 696 725 ~4

783

812 841 841 870 899

928 957 986

1015 1044 1073 1102 1131 1160 1189 1218 1247 1276 1305 1334 1363 1392 1421 1450 1479 1508 1537 1566 1595 1624 1653 1682 1682 1711 1740 1769 17'98 1827 1856 1885 1914 1943 1972 2001 2030 2059

70 71 72 73 74

2088

2117

P(ll

31 31 31 31 31 31 31 31 31 31 31 31 31 31 31 31 31 31 31 31 31 31 31 31 31 31 31 31 31 31 31 31 31 31 31 31 31 31 31 31 31 31 31 31 31 31 31 31 31 31 31 31 31 31 31 31 31 31 31 31 31 31 31 31 31 31 31 31 31 31 31 31 31 31 31

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 Z3 24 25 26 27 28 29

30 31 32

0(1, Jl 31 62 93 124 155 186 217 248 279 310 341 372 403

434 465 496 527 558 589 620

651 682 713 744 775 806

837 868 899 930

961 961

992

34 35

1023 1054 1085 1116 1147 1178 1209 1240 1271 1302 1m 1364 1395 1426 1457 1488 1519 1550 1581 1612 1643 1674 1705 1736 1767 17'98 1829 1860 1891 1922 1922 1953 1984 2015

37 38

39 40 41 42 43 44 45 46

47 48 49 50 51 52 53 54 55 56 57

59 60 61 62 63

64 65

66 67 68 69 70 71 72 73 74 ~

2046

2077 2108 2139 2170 2201 2232 2263

, P(ll

37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 Z3 24 25 26 27 28

29 30 31 32 33

34 35 36 37 38 39 40 41 42 43 44 45 46

47 48 49 50 51 52 53 54 55 56 57

59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 ~

1째(l,Jl

37 74 111 148 185 222 259 296 m 370 407 444

481 518 555 592 6Z9 666

703

740

814 851 888

925 962 999

1036 1073 1110 1147 1184 1221 1258 1295 1332 1369 1369 1406 1443 1480 1517 1554 1591 1628 1665 1702 1739 1776 1813 1850 1887 1924 1961 1998 2035 2072 2109 2146 2183 2220 2257 2294

2331 2368

2405 2442 2479 2516 2553 2590 2627 2664 2701 2738

P(ll

41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 4141 41 41 41 41 41 41 41 41 41 41 41

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29

30 31 32 33

34 35 36 37 38

39 40 41 42 43 44 45 46

47 48 49 50 51 52 53 54 55 56 57 58 59 60

61 62 63 64 65 66 67 68 69 70

71 72 73 74 ~

O(I,Jl

41 82 lZ3 164 205 246 287 328 369 410 451 492 533 574 615 656 697 738 779 820

861 902 943 984 1025 1066 1107 1148 1189 1230 1271 1312 1353 1394 1435 1476 1517 1558 1599 1640 1681 1681 1722 1763 1804 1845 1886 1927 1968 2009 2050 2091 2132 2173 2214 2255 2296 2337 Z378 2419 2460 2501 2542 2583 2624 2665 2706 2747 2788 2829 2870 2911 2952 2993 3034

The s.&randache Function S(n)zO(I,J) for powers of primes P(I)路J. PO)

59 59 59 59 59 59 59 59 59 59 59 59 59 59 59 59 59 59 59 59 59 59 59 59 59 59 59 59 59 59 59 59 59 59 59 59 59 59 59 59 59 59 59 59 59 59 59 59 59 59 59 59 59 59 59 59 59 59 59 59 59 59 59 59 59 59 59 59

59 59 59

59 59 59 59

1 2 :5 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44

45 46 47 48 49 50 51 52 53 54 55 56 57 58

59 60 61 62 63 64 65 66 67 68 69 70 71 n 73

74 75

Page :5 of 3.

0(1, J) 59 118 177 236

295 354 413 4n 531 590 649 708 767 826 885 944

1003 1062 1121 1180 1239 1298 1357 1416 1475 1534 1593 1652 1711 1770 1829 1888 1947 2006 2065 2124 2183 2242 2301 2360

2419 2478 2537 2596 2655 2714 2m 2832 2891 2950 3009 3068 3127 3186 3245 3304 3363

3422 3481 3481 3540 3599 3658 3717 3776 3835 3894

3953 4012 4071 4130 4189 4248 4307 4366

PO)

61 61 61 61 61 61 61 61 61 61 61 61 61 61 61 61 61 61 61 61 61 61 61 61 61 61 61 61 61 61 61 61 61 61 61 61 61 61 61 61 61 61 61 61 61 61 61 61 61 61 61 61 61 61 61 61 61 61 61 61 61 61 61 61 61 61 61 61 61 61 61 61 61 61 61

O(l,J)

1 2 :5 4 5 6 7 8 9 10

61 122 183 244 305 366 427

12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44

45 46 47 48

49 50 51 52 53 54 55 56 57 58

59 60 61 62

63 64 65 66 67 68 69 70 71

n 73

74 75

488

549 610 671 732 793

854 915 976 1037 1098 1159 1220 1281 1342 1403 1464 1525 1586 1647 1708 1769 1830 1891 1952 2013 2074 2135 2196 2257 2318 2379 2440 2501 2562

2623

2684 2745 2806 2867 2928 2989

3050 3111 31n 3233 3294 3355 3416 3477 3538 3599 3660

3nl 3nl 3782

3843 3904 3965 4026 4087 4148 4209 4270 4331 4392 4453 4514

P(I)

67 67 67 67 67 67 67 67 67 67 67 67 67 67 67 67 67 67 67 67 67 67 67 67 67 67 67 67 67 67 67 67 67 67 67 67 67 67 67 67 67 67 67 67 67 67 67 67 67 67 67 67 67 67 67 67 67 67 67 67 67 67 67 67 67 67 67 67 67 67 67 67 67 67 67

0(1, J)

1 2 :5 4 5 6 7 8 9 10

67 134 201 268 335 402

12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44

45 46 47 48

469

536 603

670 737 804

871 938

1005

10n 1139 1206 1273 1340 1407 1474 1541 1608 1675 1742 1809 1876 1943 2010 2077 2144 2211 2278 2345 2412 2479 2546 2613 2680 2747 2814 2881 2948 3015 3082 3149 3216

49 50 51 52 53 54 55 56 57

3551 3618 3685 3752 3819

3886

59 60 61

3953

3283

3350 3417 3484

4020

PO)

71 71 71 71 71

71 71 71 71 71 71 71 71 71 71

71 71 71 71

71 71 71 71 71 71 71

71 71

71 71 71

71 71 71 71

71 71

71 71 71

71 71 71 71

71 71 71 71 71 71 71 71 71 71 71

63 64 65

4087 4154 4221 4288 4355

4422

67 68 69 70 71 n

4489 4489

71 71 71 71

74 75

4556 4623 4690

4757 4824 4891 4958

1 2 3 4 5 6 7 8 9 10 II

12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

27 28 29 30 31 32 33 34 35 36 37 38 39 40

41 42 43 44

45 46 47 48

49 50 51 52 53 54 55 56 57 58

59 60 61

63 64

71 71

71 71 71

71 71

65 66

67 68 69 70 71

74 75

O(I,J) 71 142 213 284 355 426 497 568 639 710 781 852 923 994

1065 1136 1207 1278 1349 1420 1491 1562 1633 1704 1775 1846 1917 1988 2059 2130 2201 22n 2343 2414 2485 2556 2627 2698 2769 2840 2911 2982 3053 3124 3195 3266 3337 3408 3479 3550 3621 3692 3763 3834

3905 3976 4047 4118 4189 4260 4331 4402

P(I)

73 73 73 73 73 73 73 73 73 73 73 73 73 73 73 73 73 73 73 73 73 73 73 73 73 73 73 73 73 73 73 73 73 73 73 73 73 73 73 73 73 73 73 73 73 73 73 73 73 73 73 73 73 73 73 73 73 73

73 73 73 73

4473

4544 4615

73 73

4686

73 73 73 73

4757 4828 4899

4970 5041 5041 5112 5183 5254

73 73

J 1 2 3 4 5 6 7 8 9 10 II

12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44

45 46 47 48 49 50 51 52 53 54 55 56 57 58

59 60 61 62

63 64 65 66

67 68 69 70 71

73 73 73

74 75

O(l,J) 73 146 219 292

365 438 511 584 657 730 803 876 949 1022 1095 1168 1241 1314 1387 1460 1533 1606 1679 1752 1825 1898 1971 2044 2117 2190 2263 2336 2409 2482 2555 2628 2701 2774 2847 2920 2993 3066 3139 3212 3285 3358 3431 3504 3577 3650 3723 3796 3869 3942 4015 4088 4161 4234 4307 4380

4453 4526 4599 46n 4745 4818 4891 4964 5037 5110 5183 5256 5329 5329 5402

'SN _ DISTR, H. Ibstedt, 930322

'The values of S(n) for n < 32000 are input from the file SN.DA T and the number of values falling into each square of a 40 x 40 matrix are counted and displayed in a graph. An interresting pattern is formed by large primes while the bottom layer mainly resulting form composite numbers requires two lines in the graph. 'Set 1= 1021 and NB = 82 on HPIIP. DEFLNG A-S DIM C(51,51) CLS :WIDTH "LPT1 :",130 LPRINT TAB(50) "The Smarandache function S(n)" LPRINT TAB(SO)

:LPRI~T

:LPRINT

LPRINT TAB (35) "Number of values of S(n) in the interval OOOy Jl S(n) < OOO(y + 1). LPRINT LPRINT TAB(4)" I y" LPRINT TAB(4)" '" K=O OPEN "SN.DAT" FOR INPUT AS #1 WHILE K<32000 INCRK INPUT #1,S 1=K\800+1 :J=S\OOO+1 C(I,J) =C(I,J) + 1 WEND CLOSE #1 FOR J = 40 TO 2 STEP -1 :LPRINT USING "###";J-1; :LPRINT • I "; FOR 1=1 TO 40 IF C(I,J) =0 THEN LPRINT SPC(3); ELSE LPRINT USING "###";C(I,J); NEXT :LPRINT NEXT LPRINT USING "###";J-1; :LPRINT "I"; J=1 :FOR 1=1 TO 39 STEP 2 LPRINT USING "###";C(I,J); :LPRINT SPC(3); NEXT :LPRINT LPRINT USING "###";J-1; :LPRINT "I"; FOR 1=2 TO 40 STEP 2 LPRINT SPC(3); :LPRINT USING "###";C(I,J); NEXT :LPRINT LPRINT TAB(S) ilL"; :FOR 1=1 TO 120 :LPRINT II_II; :~EXT :LPRINT u> x" LPRINT TAB(6); :FOR I =0 TO 39:LPRINT USING "###";1; :NEXT :LPRINT LPRINT :LPRINTTAB(5) "Intervals: 8OOx~ n < 800 (x + 1). LPRINT :LPRINT TAB(5) "SN DISTR "DATE$ LPRINT CHR$(12) END

The Smerandache function Sen)

Number of values of Sen) in the interval 800y

Sen) < 5OO(y+1).

39 38 37 36 35 34 33 32 31 30 29 28 27

n 81 76 79

82 75 75 50 81 81

~ ~

B ~

19 18 17 16 15 14 13 12

85 43 44

87 48 41

4434

78 4438

82 41 47

88 40 42

82 41 41

82 90

o o

45 45

33 26 31

45 44

10 9 8 7 6 5 4 3 2 1

31 28 30

44 44

29 27 32

20262220 19 22 25 19 96 45 51 33 29 34 22 B 23 28 19 20 16 18 B 31 29 32 23 23 ~ 26 17 22 15 18 ~ 13 18 15 14 92 46 46 96 49 47 35 30 31 24 ~ 23 24 B 17 17 19 20 19 16 14 16 16 31 12 15 13 13 98 51 47 33 36 29 26 25 25 22 21 18 B 18 36 15 18 18 29 28 16 14 26 28 B 14 23 26 20 B 95 50 45 29 37 29 21 29 B 40 ~ 20 31 33 21 27 28 33 25 25 27 35 20 22 38 29 17 33 27 27 27 26 26 33 106 52 54 33 37 63 ~ 47 49 38 35 59 32 43 49 40 38 53 34 47 46 39 41 50 34 47 44 41 38 54 35 45 44 42 39 49 40 112 57 95 67 85 70 87 69 90 69 82 n 79 78 76 78 82 73 88 71 79 74 82 79 50 n M 69 M n n 74 79 n ~ 74 82 75 85 799 637 5~ 555 532 510 495 484 478 465 457 448 442 443 437 424 423 419 411 410 688 610 565 535 521 504 493 478 471 461 455 450 437 436 429 427 421 418 414 408 85

4642

41 44

263626

25 31 29

~---------------------------------------------------------------------------------------->

1 2

Intervals: SOOx

7 8

9 10 11 12 13 14 15 16 17 18 19 20 21 22 B 24 25 26 2728 29 30 31 32 33 34 35 36 37 38 39

n < SOO(x+1).

'SNP _SORT, H.lbstedt, 930322

'This program inputs the Smarandache function S(n) for powers of primes from the file SNP.DAT sorts S(n) in ascending order and writes the result to a file SNP_ASC I

OEFLNG A-5 ClS DIM 0(42,75),KP(3150),KJ(3150),SP(3150) OPEN 路SNP.OAr FOR INPUT AS #1 FOR 1=1 TO 42 :FOR J = 1 TO 75 INCRl INPUT #1 ,K1 ,K2,K3 KP(l)=K1 :KJ(l)=K2 :SP(l)=K3

NEXT :NEXT FOR 1=1 TO l :FOR J=I+1 TO l-1 IF SP(I) >SP(J) THEN SWAP SP(J),SP(J) :SWAP KJ(I),KJ(J) :SWAP KP(I),KP(J)

NEXT :NEXT OPEN 'SNP ASC" FOR OUTPUT AS #2 FOR 1=1 TO-l PRINT #2,KP(I),KJ(I),SP(I)

NEXT CLOSE #2 PRINT "END" :ENO

'SMAR_III, H" Ibstedt, 930322

This program searches for solutions to S(fn) +s(y"n) =S(z"'n). Two parameters are set in the program: n (= NM) and the largest value of S(z"n) (= NS). x, y and z are restricted to powers of prime numbers. The input to the program is provided by the file SNP ASC, which contains S(p"'n) sorted in ascending order. DEFLNG A-P :DEFDBl X. Y,Z CLS :W1DTH "LPTl :",120 DIM D(42,75),Kl (3150),K2(3150),K3(3150) NM=3 :NS=12O :l=O OPEN "SNP ASe' FOR INPUT AS #1 WHILE NOT-EOF(l) INCR L :INPUT #l,Kl (L),K2(L),K3(L) WEND :ClOSE #1 :COUNT=O LPRINT TAB(16) "Solutions to S(x"'n) + S(y"'n) = S(z"'n) for n = "NM L?RINT TA8(16) :LPRINT

-.r---,----~--~--------~--------~------~----_r----~-

--..,-

L?RINT TA8(16) :LPRINT -, Pl , S(x"n) , S(y"n) , S(z"n) ,LPRINT TA8(16) :LPRINT

-'~--r---+---+-------~------~-------+----~--~-

-in

FOR 1= NS TO 4 STEP -1 :PRINT I IF K20) MOD NM < > 0 THEN III FOR J=I-l TO 3 STEP-l IF K2(J) MOD NM < > 0 THEN JJJ FOR K=J-l TO 2 STEP-l IF COUNT = 75 THEN GOTO KKK IF K2(K) MOD NM < > 0 THEN KKK IF Kl0)=Kl(J) OR Kl0)=Kl(K) OR Kl(J)=Kl(K) THEN KKK IF K(0) = K3(J) + K3(K) THEN INCR COUNT X=Kl (K)"'(K2(K)jNM) :Y =Kl (J)"'(K2(J)jNM) :Z=Kl (1)"'(K2(I)jNM) LPRINT TAB(16) 1"; :LPRINT USING "######";Kl (1<); LPRINT "I"; :LPRINT USING "######";Kl (J); LPRINT "I"; :LPRINT USING "######";Kl 0); LPRINT "I"; :LPRINT USING "##############";X; LPRINT "I"; :LPRINT USING "##############";Y; LPRINT "I"; :LPRINT USING "##############";Z; LPRINT "I"; :LPRINT USING "#######";K3(K); LPRINT "I"; :LPRINT USING "#######";K3(J); LPRINT "I"; :LPRINT USING "#######";K(0); LPRINT "'" END IF KKK:

NEXT JJJ: NEXT III:

NEXT L?RINT TA8(16) :LPRINT

.~I--~~--~--~--------~--------~------~----~----~LPRINT CHR$(12) PRINT "COUNT = "COUNT PRINT "END" :END

Solutions to S(x"n)

P1 2 5 7 2 5 5 5 5 7 2 5 5 3 2 3 5 3 5 7 3 2 2 5 5 2 2 2 2 5 7 5 3 3 2 2 3 5 2 2 3 5 3 5 7 7 3 2 2 7 3 5 3 5 3 3 2 3 5 5 3 5 5 2 2 2 3 2 2 2 2 2 3 3 3 3

S(y"n)

17 2 2 13 3 11 7 2 2 11 2 2 2 3 7 7 2 2 2 2 3 11 2 2 5 5 5 7 13 11 3 7 2

3 3 3 3 2 2 3 3 3 3 3 3 19 19 19 19 19 19 19 19 19 19 19 19 7 7 3 3 3 3 2 17

17 17 17 17 17 5 2 2 2 2 2 2 2 3 3 3 7 7 2 2 2 2

3 2 2 3 7 13

3 11 11 3 5 5 11 5 2 11 3 11 3 7 5 3 2 2 2 7 7 3 5 7 5 2 7 3 5 7 3 2 2 5 2

13 13 13

3 2 2 2 3 3 11 11 11 11 3 5

5 5 5 2 7

S(%"n) for n = 3 y

8 5 7 128 25 25 5 5 7 128 25 25 3 8 9 5 9

5 7 27 128 128 25 25 32

16 2 8 5 7 25 3 3 8 8 9 5 128 4 3 5 9 5 7 7 27 8 64

7 3 5 3 5 27 9 8 9 5 5 9 5 5 4 8 4 3 8 8 8 2 2 3 3 3 3

S(x"n)

17 32768 8192 13

243 11 49 8192 2048 11 1024 512 32768 2187 49 49 8192 8192 2048 2048 243 11 1024 512 125 125 625

49 13 11 81 49 8192 13 729

2048 2048 81 49 13

243 11 11 81 25 25 11 25 128 11 81 11 81 7 25 81 128 128 128 7 7 27 25 7 25 128 7 27 5 7 27 32 16 5 8

59049 59049 59049 59049 262144 262144 19683 19683 19683 19683 19683 19683 19 19 19 19 19 19 19 19 19 19 19 19 343 343 6561 6561 6561 6561 65536 17

17 17 17 17 17 17 62S

32768 32768 32768 32768 32768 16384 16384 2187 2187 2187 49 49 8192 8192 8192 4096 13 13 13 729 204a 204a 204a

243 243 11 11 11 11 81

25 25 25 25 128 7

12 15 21 24 25 25 15 15 21 24 25 25 9 12 15 15 15 15 21 21 24 24 25 25 16 16 4 12 15 21 25 9 9 12 12 15 15 24 8 9 15 15 15 21 21 21 12 20 21 9 15 9 15 21 15 12 15 15 15 15 15 15 8 12 8

9 12 12 12 4 4 9 9 9 9

S(y"n)

$(%"n)

63 63 63 63

42 39 33 33

42 42 36 33 32 32 48

45 42 42

42 42 36 36 33 33 32

32 40

40 50 42 39 33

27 42 42 39 39 36 36 27 42 39 33 33 33

27 25 25 33

25 24 33

27 33 27 21 25 27 24 24 24 21 21 21 25

21 25 24 21 21 15 21 21 16 16 15 12

58 57 57 57 57 57 57 57 57 57 57 57 57 57 57 57 57 57 57 56 56 54 54

54 54 52 51 51 51 51 51 51 51 50 48

48 48 48 48 46 46

45 45 45 42 42 42 42 42 40 39 39 39 39 36 36 36 33 33 33

33 33 33 27 25 25 25 25 24 21

Solutions to S(x"n)

P1 7 2 2 13 5 11 7 2 5 2 7 3 5 3 11 2 2 2 2 5 3 2 2 3 2 7 11 2 2 3 5 7 3 5 3 5 5 2 2 3 3 3 3 2 2 2 7 2 2 5 2 2 2 5 5 2 2 5 7 7 2 3 2 2 2 5 2 2 5 5 2 2

P2 11 17

3 2 11 2 2 3 3 7 19 5 17 17 3 11 7 3 11 3 2 3 3 2 13 2 2 7 7 7 17 3 5 13 13 2 2 3 3 2 2 5 13

3 3 7 2 5 17 2 5 13 11 2 2 5 13 11 5 3 7 7 3 3 11 2 7 11 2 7 3 5

S(y"n) • S(%"n) for n • 5

P3 5 5 5 5 3 3 3 13 13 13 13 13 13 13 13 3 3 7 7 7 7 5 5 5 5 23 23 3 11 11 11 11 11 11 11 3 3 5 5 5 5 7 7 7 19 19 19 3 3 3 3 3 3 3 17 3 3 2 2 2 3 2 5 13 3 3 3 7 7 2 5 3

7 2048 4096 13 5 11 49 8 5 64

7 81 25 81 11 8 32 4 8 25 243 64 256

81 2048 7 11 8

"53 7 81 25 81

5 25 4 16 27 81 27 27 256 64 64

7 2 2 5 32 32 128 25 5 8 8 5 7 7 4 27 64 64

2 5 32 2 5 5 4 2

8192 125 13 11 512

78125 78125 78125 78125 1594323 1594323 1594323 169 169 169 169 169 169 169 169 531441 531441 2401 2401 2401 2401 15625 15625 15625 15625 23 23 177147 121 121 121 121 121 121 121 59049 59049 3125 3125 3125 3125 343 343 343 19 19 19 19683 19683 19683 19683 19683 19683 19683

2048

125 13 11 25 81 49 7 27 27 11 128 7 11 128 7 27 5

6561 6561 32768 32768 32768 2187 8192 125 13

121 17 6561 32768 121 32768 16384 177147 59049 343 19 625

17 17 2187 121 343 177147 121 6561 16384 19683 6561 32768 13 32768 2048

343 343 343 17

2187 125 13 13 32768 2048

19683 6561 16384 2048

125 13 243 Tl.9

49 2048 625 17

729 729 729

49 49 2048

25 27

S(x"n)

S(y"n)

S(%"n)

110 85 81

145 145 145 145 135 135 135 130 130 130 130 130 130 130 130 126 126 126 126 126 126 125 125 125 125 115 115 114 110 110 110 110 110 110 110 105 lOS 105 105 lOS lOS

65 65 54

98 98 98

63 63 60 85 85

95 95 95 93

60 64

65 25 55 63

16 25 32 35 45 45 45 55 16

110 80

114

lOS 98 95

85 85 85 75

110

12 16 45 54 32 44 45

114 110 81

93 81 80

35 55 16 12 12 25 35 45 45 45 25 45 12 24

80 60 98 98 98

33 44 32 32 35 8 8 25 28 28 38

45 25 16 16 25

35 35 12

85 75

65 65 65 80 60

93 81

68 65 65 55

93 93 93 93 93

65 65 55 45 45

81 81 80 80 80

32 32 8 25 28 8

33 33

68 65 65

55 38

35 55

25 12 8

35 33

63 63 63 63 63 60

45 33

Solutions to S(x"n) • S(y"n) • S(z"n) for n • 7

Pl 3 17 2 2 3 2 2 3 2 7 3 11 3 2 7 13

7 13

3 3 2 3 5 5 2 2 2 2 3 2 11 3 3 3 3 3 3 2 2 5 3 2 3 2 2 2 2 7 2 2 3 2 2 2 2 2 2 2 2 5 2 2 5 2 3 2 3 3 3 2 2 5

11 29 5 5 13

5 5 13

5 17 5 3 13 11 31 7 5 5 5 5 5 11 13

23 23

7 7 7 43 43 43 41 41 41 41 11 19 19 19 19 19 13

11 7 5 5 19 7 5 5 11 11 11 5 11 11

13 13 13 7 37 5 5 5 17 17 17 17 5 11 11 11 11 11 7 31 29 29 29 7 7 5

5 5 3 5 11 5 17 7 13 3 3 3 3 2 3 3 2 3 2 3 2 2 2 7 3 3

13 13

7 31 29 5 11 5 23

13 23 23

3 11 11 11 19 19 7 7 3 5 17 3 7 7 13

13 5 5 3 11 2

2187 17 4 64

19683 4 64

19683 16 7 729

121 6561 64

7 13

49 13

9 81 256

6561 125 125 32 32

128 2 3 32 11 729

27 27 243 729 729

1024 1024 25 81 256

9 16 16 2 8 7 128 4 3 8 8 128 128 4 64 4 64 25 256

16 5 4 27 4 27 3 9 32 16 5

1331 29 9765625 1953125 169 9765625 1953125 169 1953125

529 529 823543 823543 823543 43 43 43 41 41 41 41 14641 361 361 361 361 361 2197 2197 2197 2197 117649 37 1953125 1953125 1953125

289

78125 59049 169 1331 31 2401 15625 15625 390625 78125 78125 121 169 169 16807 31 29 390625 1331 78125 23

121 2401 15625 3125 19 343 3125 3125 121 121 121 3125 121 121 23

3125 625

6561 3125 121 625

17 49 13

6561 729

1024 81 729

1024 243 64 243 64

1024 256

7 27 9

289 289 289 289

390625 1331 1331 1331 1331 1331 16807 31 29 29 29 2401 2401 15625 169 169 169 169

S(x"n)

S(y"n)

5(z"n)

102 119 16

220

322 322 301 301 301 301 301 301 287 287 287 287 286

132 16 46

132 32 49 87 143 117

2Q3

285

255 169 285 255 169 255 238 200

144 169

220

266

49 91 91 91 30 60 60 117

217 175 175 175

266 266 266

230

260 260 260

90 90 38 38

52 8 18 38 IT

87 45 45 75 87 87

n n

60 60

200 200

143 169 169 217 217 2Q3

230 220 200

161 143 175 175 145 133 133 145 145 143 143 143 145 143 143 161 145 120 117 145 143 120 119 91 91 117 87 117 87

266

260

259 259 255 255 255 238 238 238 238 230 220 220

220 220 220

217 217 2Q3 2Q3 2Q3

625

17 2187 49 49

32 30 16 45 16 45 18

175 175 175 169 169 169 169 161 161 144 143 143 143 133 133 133 133 132 120 119 102 91 91 91 91

729

32 30

49 45 30

11 256

23 23

59049 121 121 121 19 19 343 343 19683

13 125 125

32 32 8 24 49 52 16 18 24 24 52 52 16 46

16 46

75 46

IT 60

Solutions to S(x"n) + S(y"n) = S(z"n)

Pl 17

29 37 11 61 5 11 17 11 3 47 11 5 13 2 17 29 53 13 7 5 47 13 13 7 3 11 23 11 41 43 5 3 7 7 13 19 5 3 11 11 23 29 31 7 13 13 11 23 5 31 11 7 3 5 3 5 7 19 11 5 13 11 2 7 19 11 3 2 7 5 5 29 13 11

P2 53 47 43 31 31 29 37 19 71 23 71 67 19 17 19 47 41 29 43 43 17 59 29 19 19 19 73 71 53 53 11 67 41 59 13 73 67 13 37 71 61 59 53 11 37 11 61 53 11 17 43 43 23 11 11 61 61 37 37 31 13 29 23 13 13 43 41 59 19 43 11 41 17 23 47

P3 41 41 41 41 41 61 61 61 61 31 59 59 29 29 37 37 37 37 23 23 53 53 17 17 17 47 47 47 47 47 47 19 31 31 43 43 43 43 29 41 41 41 41 41 17 37 37 37 37 37 37 37 13 13 13 13 13 13 13 73 73 71 67 23 23 31 31 13 13 13 13 13 13 59 29

fo~

n = 11

17 29 37 ln1561

61 25 11 4913 161051 243 47 161051 5 28561 4 17 29 53 169 2401 25 47 13 169 2401 3 121 23 14641 41 43 15625 81 16807 49 13 19 15625 243 11 121 23 29 31 7 13 13 121 23 15625 31 1331 7 729 125 729 125 117649 361 11 15625 13 121 32

343 19 121 3 64 343 3125 3125 29 13 11

2809 2209 1849 961 961 70nal 50653 130321 71 6436343 71 67 47045881 83521 47045881 2209 1681 841 1849 1849 24137569 59 24389 130321 130321 2476099 73 71

53 53 161051 67 1681 59 4826809

73 67 371293 1369 71 61 59 53 161051 1369 ln1561

61 53 161051 4913 43 43 12167 ln1561 ln1561

61 61 37 37 961 2.8561 841 529 371293 28561 43 41 59 6859 43 14641 41 289 529 47

68921 68921 68921 68921 68921 ml 3n1 3nl 3nl 923521 3481 3481 707281 7On81

50653 50653 50653 50653 6436343 6436343 2809 2809

24137569 24137569 24137569 2209 2209 2209 2209 2209 2209

2476099 29791 29791 1849 1849 1849 1849 24389 1681 1681 1681 1681 1681 1419857 1369 1369 1369 1369 1369 1369 1369 4826809 4826809

4826809 4826809

4826809 4826809 4826809

73 73 71 67 12167 12167 961 961 371293 371293 371293 371293 371293 371293 59 841

S(x"n)

S(y"n)

S(z"n)

187 319 407 671 671

1166 1034

1247 1221

1353 1353 1353 1353 1353 1342 1342 1342 1342 1333 1298 1298 1247 1247 1221 1221 1221 1221 1219 1219 1166 1166 1071 1071 1071 1034 1034 1034 1034 1034 1034 1007 992

121 544 561 114 517 561 50 533 24 187 319 583 273 273 95

517 143 273 273 27 231 253 451 451 473 270 90 343 140 143 209 270 114 121 231 253 319 341 70 143 143 231 253 270 341 341 70 135 135 135 135 399 399 121 270 143 231 60

203 209

231 27 68 203

225 225

319 143 121

946 682 682 798

781 1219 781 737 1197 714 1197 1034 902 638

946 946

1071 649 928 798 798

1007 803

781 583

583 561 737 902 649 806 803

737 676 814 781 671 649 583 561 814 671 671 583 561 544

473 473 736

671 671 671 671 407 407 682

533 638

506 676 533 473 451 649 608

473 451 451 357 506 517

99Z 946 946 946 946

9Z8

902 902 902 902 902 884

a14 814 814 814 a14 814 814 806 806 806 806 806 806 806

803 803 781 737 736 736 682 682

676 676 676 676 676 676 649 638

'SMAR _iv, H. Ibstedt, 930323

'This program searches for solutions to the equation S(k"nri = S(n"'k). The search is limited to the first 8000 values of S(n) loaded from the file SN.DAT. No non-trivial solutions were found. DEFLNG I-S :DEFDBL X,Y,Z CLS :WIDTH "LPT1:",120 DIM S(8000) L=O :NS=90 :KS=90 OPEN "SN.DAT" FOR INPUT AS #1 FOR L=1 TO 8000 INPUT #1 ,S(L)

NEXT CLOSE #1 PRINT S(8000) LPRINT TAB(12) "Search for solutions to S(\("n)"'i = S(n"'k). "

---r------r---..,..

lPR I NT TAB( 12) : lPR I NT "r"1

--...,.....----r---.,..----,I"

"I Ie n S(Ie"n)1 n Ie S(n"Ie)1 I" "I-I--+---+--~I---I--_4--~II---~I"

lPRINT TAB(12) :lPRINT lPRINT TAB( 12) : lPRINT

FOR K=2 TO KS :PRINT K FOR N=2 TO NS IF K"'N>8000 OR N"K> 8000 THEN N=NS :GOTO L2 C=S(K"'N) :D=S(N"K) IF C>D THEN SWAP C,D E=1 :1=0

L1: E=E*C :1=1+1 IF E>D THEN L2 IF E=D AND K< >N THEN GOSUB LW IF E<D THEN L1 L2:

NEXT NEXT ,,~,--'"---......1.---'------1.---'---'------',,,

lPR I NT TAB( 12) : lPR I NT

LPRINT CHR$(12) PRINT "END" :END LW: LPRINT TAB(12) 1"; :LPRINT USING "######";K; LPRINT" I "; :LPRINT USING "######";N; LPRINT" I"; :LPRINT USING "######";S(K"N); LPRINT "I"; :LPRINT USING "######";N; LPRINT" I"; :LPRINT USING "######";K; LPRINT "I "; :LPRINT USING "######";S(N"K); LPRINT "I"; :LPRINT USING "#######";1; LPRINT "I" RETURN Search for solutions to S(Ie"n)"i Ie

S(Ie"n)

n 2 4

4 2

6 6

= S(n"Ie)" S(n"le)

Ie 4 2

2 4

6 6

i 1 1

'SMAR_ Y, H. Ibstedt, 930323

This program uses the first 32000 values of S(n) input from the file SN.DAT to calculate 1+ 1/S(2) + 1/S(3) + ... + 1/S(n) for n=800, 1600, 2400, ... 32000. The sums are used to study the behaviour of 1+ 1/S(2) + 1/S(3) + ... 1/S(n)-T(n). The following cases are examined T(n) =0, T(n) = Log(S(n)), T(n) equal to the logarithm of the largest prime less than n and finally T(n) = (rK-V) + 1/2"a+ 1/3"'a+ ... + 1/n"a, where a=.5164, r=0.96 and V=27. K=n/800. 'Set 1=22 and NB=62 on HPIIP 'Number of intervals = NN (40), Size of interval NI (800). DEFDBL A-Z :DIM LAM(40),SAM(40),ZAM(40),HAM(40) NN=40 :NI=800 CLS :WIDTH "LPT1 :",120 Z=1 :K=1 :Z1 =0 LPRINT :LPRINT LPRINTTAB(12); "Behaviour of 1 +1/S(2)+1/S(3)+ ... 1/S(n)-T(n). K=n\800." LPRINT TAB(12); "In the column LAM(K) T(n) =0, in SAM(K) T(n) = Log(S(n)),in" LPRINT TAB(12); "ZAM(K) T(n) is the logarithm of the largest prime less路 LPRINTTAB(12); "than n and in HAM(K) T(n)=1 +1/2".52+1/3".52+ ... 1/n".52'" :LPRINT LPRINT TAB( 12); "r-I---...,.----,..------r"---~--__,I" LPRINT TAB(12}; LPRINT TAB(12);

"I K LAM(K) SAM(K) ZAH(K) HAM(K) I" "t-I----+I----+----+----!------tl"

OPEN "SN.DAr FOR INPUT AS #1 WHILE K<41 G=1 WHILE 1< K*NI INCRI INPUT #1,S IF S>G THEN G=S Z=Z+1/S Z1 =Z1 +1/1".5164 WEND LAM(K)=Z HAM(K) =Z-Z1-.96*K+ V SAM(K) =Z-LOG(S) ZAM(K) =Z-LOG(G) LPRINT TAB(12); :LPRINT 1"; :LPRINT USING "##########";K; LPRINT "I"; :LPRINT USING "##########";LAM(K); LPRINT "I'; :LPRINT USING '##########";SAM(K); LPRINT "I"; :LPRINT USING "##########";ZAM(K); LPRINT" , "; :LPRINT USING "##########";HAM(K); LPRINT "I" INCRK WEND CLOSE #1 LPRI NT TAB( 12);

"I-'_ _ _........_ _ _"--_ _--'-_ _ _.........._ _---'1"

LPRINT :LPRINT TAB(12) 'SMAR _v" LPRINT CHR$(12) PRINT "END" :END

PCW, Number'$ ~unt: Problem.M. February 1393

S 35

91 102 112 121 130 138 148 154 162 169 176 183 190 197 203 210

K 21

S 216

223

228

235

6 26

7 27

8 28

9 29

30 31

12 32

13 33

14 34

15 35

16 36

19 20

37 38 39 40

240 248 252 258 262 268 273 279 285 289 295 300 304 310 315 320

1 +1/S(2)+ ... +1/S(n)-log(S(n» n = 800, 1600, 2400, ... 32000 350 300 250 ........

C/')

---<U

200

E E 150 <U

C/')

100 50 O~~~~~~~~~±P~~~~~~~~~P*~~~

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40

K = n/800

Behaviour of 1+1/S(2)+1/S(3)+ ... l/S(n)-T(n). K=n\800. In the column LAM(K) T(n)=O, in SAM(K) T(n)=Log(S(n»,in ZAM(K) T(n) is the logarithm of the largest prime less than n and in HAM(K) T(n)=(r*K-V)+1/2-a+1/3-a+ ••. l/n-a, where r=0.96, V=27 and a=0.5164 have been chosen to fit HAM(K) as closely as possible to O. K

LAM(K) 1 2 3 4 5 6 7 8 9 10 11

SMAR v

12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40

37 55 70 82 94 104 114 123 132 141 149 157 164 172 179 186 193 200 206 212 219 225 231 237 243 249 254 260 265 271 276 282 287 292 297 303 307 313 317 322

ZAM(K)

SAM(K) 35 53 67 80 91 102 112 121 130 138 146 154 162 169 176 183 190 197 203 210 216 223 228 235 240 246 252 258 262 268 273 279 285 289 295 300 304 310 315 320

31 48 62 74 85 96 105 115 123 132 140 148 155 162 170 176 183 190 196 203 209 215 221 227 233 239 244 250 255 261 266 272 277 282 287 292 297 302 307 312

HAM(K) 13 9 6 5 3 2 1 1 0 -0 -0 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -0 -0 -0 -0 -0 0 0 0

The Smarandache Function S(n)

Graphical representation of 1 + 1/5(2) + 1/5(3) + ... + 1/5(n) - T(n) for n < 32000. Graph I: Graph II: Graph III: Graph IV:

T(n) =0. T(n) = log(5(n)). T(n)=logOargest prime<n). T(n) = (rK-V) + 1/2"'a + 1/3"'a+ ... + 1/n"'a, where the parameters r=O.96, V =27 and a = 0.5164 were chosen to fit the graph as closely as possible to a horizontal line.

MO~~~~~~~~~~~~~~~ 1+ 1/S(2) + 1/S(3) + ... + 1/S(n)-T(n) 250

150 100 50

135

2 4 6

9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40

I -+- II -

III -+- IV

'SMARAND1, H. Ibstedt, 930327

The Smarandache function S(n) calculated by comparing largest prime and S(P"A). The upper limit for the calculation is n = 1000000. The results are used to calculate 1 + 1/S(2) + 1/S(3) + ... + 1/S(n) registering partial sums for n = 25000, 50000, 75000, ... 1000000. DEFLNG A-S CLS :T=TIMER DIM P(168),D(168,20),K(168),L(168) OPEN "PAM FOR INPUT AS #1 FOR 1=1 TO 168 :INPUT #1,P(I) :NEXT :CLOSE #1

This part of the program calculates S(P(/)"'A) and saves the result in the array D(/,A), P(/) is the /th prime number. The routine uses the fact that D(/,A) < =P(/)*A in the search for the value of D(/,A). This calculation goes as far as is required to calculate S(n) up to n = 1000000. FOR 1=1 TO 168 A=2 :P=P(I) :0(1,1) =P WHILEA<21 C=O :N=O L: INCRC INCR N,P IF C> =A THEN D(I,A) = N :GOTO LWEND pp=p*p l1: IF N-PP*INT(N/PP) =0 THEN INCR C :pp=pp*p :GOTO l1 IF C> =A THEN D(I,A) =N :GOTO LWEND :ELSE L LWEND: INCRA WEND NEXT

This part of the program calculates S(N) and the sum of reciprocals. It calls on the subroutine NFACT to express N in prime factor form. Factors P(/)"'A with A> 1 are replaced by D(/,A) and placed in array LO together with the factors P(I) of multiplicity 1. S(N) is then the largest component of LO. S(N) is stored in a file SN.DAT. N=1 :Z=1 :0=0 :ZC=O OPEN "SAM.DAT" FOR APPEND AS #3 WHILE N<1000001 if inkey$< > •• then print "end" :end INCR N :INCR 0 :PRINT N 'Factorize N. GOSUB NFACT IF K(O) >0 THEN S=P(O) :GOTO LWR 'Construct LO. FOR 1=1 TO 168 :L(I)=O :NEXT C=O FOR 1=1 TO M INCRC 68

IF K(I) = 1 THEN L(C) = P(I) IF K(I) > 1 THEN L(C) = D(I.K(I)) NEXT

'Find the largest value of LO and hence SeN). S=O FOR 1=1 TO C IF L(I) > S THEN S = L(I) NEXT LWR: Z=Z+1/S IF 0=25000 THEN ZC=ZC+Z :WRITE #3.Z.ZC :Z=O :0=0 WEND CLOSE #3 T = TIMER-T :PRINT T END

'Subroutine for factorizadon of N. (Improved from SMARAND to avoid large primes) NFACT: FOR 1=0 TO 168 :K(I)=O :NEXT :P(O)=0 N1 =N :1=0 :M=O FOR 1=1 TO 168 LA: IF N1-P(I)*fNT(N1/P(I))=0 THEN K(I)=K(I)+1 :M=I :N1 =N1/P(I) :GOTO LA IF N1 =1 THEN 1=168 NEXT IF N1 > 1 THEN P(O) = N1 :K(O) = 1 RETURN

Problem v: Summary of obtained data.

Sum interval

Graph I Sum 1/S(n)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33

277 145 119 105

277 422 541

46 46

742 831 914 993 1069 1142 1212 1280 1346 1410 1473 1534 1594 1652 1709 1765 1820 1874 1927 1979 2031 2082 2132 2182 2231 2280 2328 2375 2422 2468 2514 2559 2604 2648 2692 2736

35 36 37 38 39 40

89 83 79 76 73 70 68 66

63 61 60 58 57 56

55 54 53

52 52 51 50 50 49 49 48 47 47

45 45 44 44 44

646

Graph II Sum 1/n"'2

Diff.

315

-38

446

-24 -5 25 36 58 79 100 122 143 165 186 207 228 250 271 292 312 332 352 372 392 412 431 451 471 490 510

546

621 706 n3 835

893 947 999 1047 1094 1139 1182 1223 1263 1302 1340 13n 1413 1448 1482 1515 1548 1580 1611 1642 1672 1701 1731 1759 1787 1815 1842 1869 1896 1922 1948 1973 1999

530

549 569 588

607 626 645 663

682 700 719 737

Graph III after Un. Carr. lin. carr. -38 -18 2 22 42 62 82 102 122 142 162 182 202 222

242 262 282 302 322 342 362 382 402 422 442 462 482 502 522 542 562 582 602 622 642 662 682 702 722 742

0 -6

-7 3 -6 -4 -3

-2 0 1 3 4 5 6 8 9 10 10 10 10 10 10 10 9 9 9 8 8 8 7 7 6 5 4 3 1 0 -2 -3 -5

The Smarandache Function S(n) Comparison with the sum of 1In A 0.5

for n up to 1000000 3000Tr=============~------------------------~

II: 1 +1/8(2)+ .. +1/S(n) I

2500

II: 1 +1/S(2)+ .. +1/S(n)-(1 +1/2" .5+ .. +1/n" .5)

2000+-----------------------~~--------------~

1500+--------1.~--~~------------------------~

1000+-------~--------------------------------~

500~~----~~------~~~~~~~====~

IK=n/25000 I

III: After linear correction of II by (20K-58) -500~~~~~~~~~~~~~~~~~~~~~~

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 K 2 4 6 8 10121416182022242628303234363840

CON TEN T S

Dr. Constantin Dumitrescu, A brief HISTORY of the "Smarandache Function" . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 ~ke Mudge, The Smarandache Function, together with a sample of The Infinity of Unsolved Problems associated with it .... 10 ~ke Mudge, Review, July 1992, The Smarandache Function: a first visi t ? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Jim Duncan, Algorithm in Lattice C to generate S(n) ......... 11 Jim Duncan, Monotonic Increasing and Decreasing Sequences of S(n) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Jim Duncan, On the Conjecture D/ k ) (1) =1 or 0 for k>=2 ........ 17 John C. McCarthy, A Simple Algorithm to Calculate S(n) ...... 19 ~ke Mudge pays a return visit to the Florentin Smarandache Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ~ke Mudge, Review of Numbers Count -118- February 1993: a revisit to the Florentin Smarandache Function . . . . . . . . . . . . . . 32 Pal Gr{2Jnas, A Note on S(pr) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 Pal Gr{2Jnas, A Proof of the Non-Existence of "Samma" ......... 34 John Sutton, A BASIC PROCedure to calculate S(pAi) .......... 36 Henry Ibstedt, The Florentin Smarandache Function Sen) -programs, tables, graphs, comments . . . . . . . . . . . . . . . . . . . . . . 38-71

$ 9.95

A collection of papers concerning Saarandache type functions, numbers, sequences, integer algorithms, paradoxes, experiaenta1 geometries, algebraic structures, neutrosophic probability, set, and logic, etc. Dr. C. Dum.i. trescu & Dr. V. Seleacu Department of Mathematics Universi ty of Craiova, Roman j a;