An Illustration of the Distribution of the Smarandache Function
by Henry Ibstedt The cover illustration is a representation of the values of the Smarandache function for 1]:5; 53. The group at the back of the diagram essentially corresponds to S(P) =p, the middle group to S(2p) =P (pf2) while the front group represents all the other values of S( n) for 1]:5; 53. Diagram 1. Distribution of S(n) up to n
= 32000
(not to scale)
y Number of values of S(n) on lines y= kx .
Number of values of S(n) falling between lines:
1862
707
22835 values of S(n) fall below the line y = xJ6
x
It may be interesting to take this graphical presentation a bit further. All the values of S(n) for 1]:5;32000 (conveniently chosen in order to use short integers only) have been sorted as shown in table 1. Of the 19114 points (o,S(nÂť situated above the line y = x/50 only 61 points fall between lines. All of these of course correspond to cases where n is not square free. Diagram 1 illustrates this for the lines y=x, y=x/2, y=x/3, y=x/4, y=x/5 and y=x/6. The top line contains 3433 points (o,S(n) although there are only 3432 primes less than 32000. This is because (4,S( 4Âť belongs to this line. 34
TABLE 1. On the distribution of the Smarandache runction Sen) for ns32000. N = number of values of Sen) on the line y=x/k, i.e. S(n)=n/k. The points are the only ones between lines y=x/k and y=x/(k+l) for k<50. (n, S (nÂť k
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
3433 1862 1302 1006 832 707 616 550 495 450 417 387 359 336 321 301 283 273 256 250 239 227 213 218 204 196 190 187 176 179 163 164 159 154 154 153 139 139 136 139 125 133 119 125 126 117 109 120 114 112
Points (n,S(n) ) between lines: 9, 6) 16, 6) 49, 14) 121, 169, 45, 125, 361, 147, 529, 80, 841, 961, 250, ( 175, (1369, ( 507, ( 243, (1849, ( 225, (2209,
22) 26) 6) 15) 38) 14) 46)
75, 10) 289, 34)
6) 58) 62) 343, 21) ( 363, 22) 15) 245, 14) 10) 74) 26) 12) (1681, 82) 86) 10) 94)
( 256, 10) (2809,106) ( 605, 22) (1083, 38) (3481,118) (3721,122) ( 441, 14) ( 686, 21) ( 500, 15) (1587, 46) (5041,142) (5329,146) ( 539, 14) (6241,158) ( 486,12) (6889,166) ( 512, 12) (2523, 58) (7921,178) ( 637, 14) (2883, 62) (1805, 38) ( 729, 15) (1089, 22)
Number of elements below y
25, 10)
x/50:
( 867, 34)
( 625, 20) ( 845, 26) (4489,134)
( 847, 22) (1331, 33) (1445, 34) (1183, 26) (9409,194)
12774 .
35