Smarandache k-k additive relationships by Ryan Elias

Smarandache k-k additive relationships Henry Ibstedt

Abstract: An empirical study of Smarandache k-k additive relationships and related data is tabulated and analyzed. It leads to the conclusion that the number of Smarandache 2-2 additive relations is infinite. It is also shown that Smarandache k-k relations exist for large values ofk. We recall the definition of the Smarandache function Sen): Definition: S(nJ is the smallest integer such that S(n)! is divisible by n. The sequence of function values starts: n:

Sen):

1 0

2 2

5 5

6 3

7 7

8 4

9 6

10 5

A table of values of Sen) up to n=4800 is found in Vol. 2-3 of the Smarandache Function Journal [1]. Definition: A sequence offunction values S(n), S(n+1)+ ... +S(n+2k-1) satisfies a k-k additive relationship if S(n)+S(n+ 1)+ ... +S(n+k-l)=S(n+k)+S(n+k+ 1)+ ... +S(n+ 2k-l) or k-J

2k-J

LS(n+ j) = LS(n+ j) j=0

j=k

A general definition of Smarandache p-q relationships is given by M. Bencze in Vol. 11 of the Smarandache Notions Journal [2]. Bencze gives the following examples of Smarandache 2-2 additive relationships: S(n)+S(n+l)=S(n+2)+S(n+3) S(6)+S(7)=S(8)+S(9), 3+7=4+6; S(7)+S(8)=S(9)+S(10), 7+4=6+5; S(28)+S(29)=S(30)+S(31), 7+29=5+31. He asks for others and questions whether there is a finite or infinite number of them. Actually the fourth one is quite far off: S(114)+S(115)=S(116)+S(117),19+23=29+13; The fifth one is even further away: S(1720)+S(I721)=S(1722)+S(I723), 43+1721=41+ 1723. It is interesting to note that this solution is composed to two pairs of prime twins (1721,1723) and (43,41), - one ascending and one descending pair. This is also the case with the third solution found by Bencze. One example of a Srnarandache 3-3 additive relationship is given in the above mentioned article: S(5)+S(6)+S(7)=S(8)+S(9)+S(1 0), 5+3+7=4+6+5. Also in this case the next solution is far away: S(5182)+S(5 I 83)+S(5184)= S(5185)+S(5186)+S(5187), 2591 +73+9=61 +2593+ 19. 62

To throw some light on these types of relationships an online program for calculation of Sen) [3] was used to tabulate Smarandache k-k additive relationships. Initially the following search limits were set: ~107; 2~6. For k=2 the search was extended to 8 ~1 0 â&#x20AC;˘ The number of solutions m found in each case is given in table 1 and is displayed graphically in diagram 1 for 3~6. The numerical results for ~ are presented in tables 4 -8. Table 1. The nwnber m ofSmarandache k-k additive solutions for n<107. k

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

158

8 11

m 45

35 30 25 20 15 10 5 O. <')

It')

,...

(])

.... ....

....

<')

....

It')

....,...

Diagram 1. The nwnber m ofSmarandache k-k additive relationships for n<107 for 3~6.

The first surprising observation - at least to the author of these lines - is that the number of solutions does not drop off radically as we increase k. In fact there are as many 23-23 additive relationships as there are have 10-10 additive relationships and more than the number of 8-8 relations in the search area n<107. The explanation obviously lies in the distribution of the Smarandache function values, which up n=32000 is displayed in numerical form on page 56 of the Smarandache Function Journal, vol. 2-3 [1]. This study has been extended to ~107. The result is shown in table 2 and graphically displayed in diagram 2 where the number of values Z ofS(n) in the intervals 50000Oy+l::::;S(n)~00000(y+1) is represented for each interval 500000x+l~00000(x+l) for y=0,1,2, ... ,19 and x=0,1,2, ... ,19.The fact that S(P)=p for p prime manifests itself in the line of isolated bars sticking up along the diagonal of the base ofthe diagram. The next line, which has a gradient = 0.5 ,corresponds to the fact that S(2p)=p. Of course, also the blank squares in the base of the diagram would be filled for n sufficiently large. For the most part, however, the values of Sen) are small compared to n. This corresponds to the large wall running at the back of the diagram. A certain value of Sen) may be repeated a great many times in a given interval. For n<10 7 82% of all values of n correspond to values of Sen) which are smaller than 500000. It is the occurrence of a great number of values of Sen) which are small compared to n that facilitates the occurrence of equal sums of function 63

values when sequences of consecutive values of n are considered. If this argument is as important as I think it is then chances are good that it might be possible to find, say, a Smarandache 50-50 additive relationship. I tried it - there are five of them, see table 9. Of the 158 solutions to the 2-2 additive relationships 22 are composed of pairs of prime twins. These are marked by * in table 3. Of course there must be one ascending and one descending pair, as in 9369317+199=9369319+197 A closer look at the 2-2 additive relationships reveals that only the first two contain composite numbers. Question 1: For a given prime twin pair (p,p+2) what are the chances that p+l has a prime factor q 2 such that q+2 is a factor ofp-l or q-2 a factor ofP+3? Question 2: What percentage of such prime twin pairs satisfy the Smarandache 2-2 additive relationship? Question 3: Are all the Smarandache 2-2 additive relationships for n>7 entirely composed of primes?

To elucidate these questions a bit further this empirical study was extended in the following directions. 8

1. All Smarandache 2-2 additive relations up to 10 were calculated. There are 481 of which 65 are formed by pairs of prime twins. 2. All Smarandache function values involved in these 2-2 additive relationships for 7<~1 08 were prime tested. They are all primes. 8 3. An analysis of how many of the Smarandache function values for n<10 are primes, even composite numbers or odd composite numbers respectively was carried out.

The results of this extended search are summarized by intervals in table 3 from which we can make the following observations. The number of composite values of Sen), even as well as odd, are relatively few and decreasing. In the last interval (table 3) there are only 1996 odd composite values. Even so we know that there are infinitely many composite values of Sen), examples S(p2)=2p, S(P3)=3p for infinitely may primes p. Nevertheless the scarcity of composite values of Sen) explains why all the 2-2 additive relations examined for n>7 are composite. The number of 2-2 additive relations is of the order of 0.1 % of the number of prime twins. The 2-2 additive relations formed by pairs of prime twins is about l3.5% of the prime twins in the respective intervals. Although one has to remember that we are still only "surfing on the ocean of numbers" the following conjecture seems safe to make: Conjecture: The number of Smarandache 2-2 additive relationships is infinite. What about k>2? Do k-k additive relations exist for all k? If not - which is the largest possible value ofk? When they exist, is the number ofthem infinite or not?

20 19

31089

31342

31516

31613

31891

31908

32049

32287

32565

16271

32802

16437

32996

16567

33334

33744

16921 17148

34139

17453

34778

11470

11350

11319

11604

11533

8730

8723

8683

16365

16429 11328 15614

65130 65604 65992

11250

11166

8588

8560

7014

6931

11153

11150

8497

8494

136556

12788

12714

1855.11 270551

88971 101232

17971

17686

12033

11852

20793

8950

16060

16066

13102

13119

18125

11059

15515

15488

13592

13483

18700

30791

21798

28891

22955

28086

23553

27681

23970

27206

24323

26997.

24500

26864

24601

26650

24762

26716

495999

463040

445643

434431

426092

419679 414225 409741

405704

402172

399158

396323

393706 391352

389193

387190

385253

383470 381848

.180148

8208367

35657 499999

16573

16823

16991 11641

16761

16294

36960

:3 2

17325

31001 31370

31342

Sum

31089

31370

31001

----_._--

Table 2. The number of values z of S(n) in the intervals SOOOOOy+ 1:5;8(n):5;SOOOOO(y+ 1) is represented for each interval SOOOOOx+ 1SnSSOOOOO(x+ 1) for y=O, I,2, ... ,19

and x=O,1,2, ... ,19.

II) ~

Distribution of S(n)

..-

Diagram 2. The distribution ofS(n) for n<107.

Table 3. Comparison between 2-2 additive relations and other relevant data. Interval

n ~107

58980

158

9932747

48427

9957779

38023

4198

45485 43861 42348 41547 40908 39984 39640 39222 440402

37 42 40 30 28 41 20 26 481

4 4 5 2 4 7 4 4 65

9963674 9967080 9969366 9971043 9972374 9973482 9974414 9975179 99657140

32922 29960 27962 26473 25303 24327 23521 22825 310355

3404 2960 2672 2484 2323 2191 2065 1996 99999999

.0\ - 10

7 < n ~ 5 -107

5_107 <ns6-107 6·107 <nS7.107 7_107 <ns8-107

8_107 <ns9-107 9-107 <n<108 Total

formed *of twins pairs

•

107 <n~2·107 2-10 7 <nS3-10 7 3-10 7 <nS.o\-10 7

i of 2-2 additive relations

t of S. function primes

of S. function even values 59037

prime twins

of S. odd composite values 8215

Table 4. Smarandache function: 2-2 additive quadruplets for n<107 t

Sen)

S(n+1)

S(n+2)

8 (n+3)

1 2

6 7

7 4

4 6

3 4

3 7 7

114

23 1721

5 29

6 5 31

1720

19 43

6 7

3538 4313

61 227

3539 719

8 9

8474

223

10300

103

13052

10 11

15417

15688

19902

14 15

22194

113 10301

863 163

3541 83 173

101

10303

251 571

229

107

593

907

373 257

53 107

541

523

1531

311

71 1327

137 577

193 97

179 643

151 31

197

241 281

107 587

331 127

16 17

22503 24822 26413

18 19

56349 70964

20 21

75601 78610

22 23 24 25

86505 104309

79 104309

107083 108016

6299 157

26 27

108845 125411

1979

130254 133455 147963

877 1277

29 30

433 2087 157 173 1123

31 32 33

171794 187369

43 1753 71

189565

1223

191289

35 36 37

198202 232086 247337

9109 877

269124 286080

13 1723

1523 137

6047 167

6551 157

104311

1409 1187 6047

59 353 1223

6803

1493

1511

859

239 439

1163 421 293

353 59 271

1481 191 59

1153 337 1481

8317 199

809

521 881 457 317 47 131

619 89 59 7649

433 247339

547

2153 547

941 457

541 149

113 103 2281 151 673

983 631 109 3881

733

103 20341 191

3169 4967 479029

1759 239 41

39 40 41

323405 330011

42 43 44

342149 403622 407164

421474

46 47

427159 479026

193

181 479027

48 49

497809

257

743

227

773

526879

50 51

539561 564029

12253 271 179

89 4733

10331 1867

2089

1009

2011 3137 1259

598517 603597

811 3391 2213 557

109 4051 281 79

109

618031

53 54 55

604148 604901

618029

2539 25127

449 1163 2069 433 618029

859

839

151

6959 743

991

151

911 1579 79

103 461

823 247337 149

587

23 83 367

1051 2251 3229 557

1151 503 4001 911 107

Table 4. ctd t

8 (11)

8(11+1)

8 (11+2)

8(11+3)

57 58 59 60 61 62 63 64

662383 665574

4219 53 229

41399 337 675551

44159 307 227

1459 83 675553

313 59 4073 1583

681089 2339

681091 1907 1583

2137 1831

283 181

311 491 2887 151 151 691

797 1151 10259 2663

101 881 499 43

41 1553 157 2273

857 479 10601

2269

277

163

103 1114271

1008989 73

101 1114273

2383 1008991 71

5641 829 239 2927

4099 5101 1039 3517

109

9631

1198734 1316412 1343493 1353260 1362471 1382345 1397236 1457061

953 53 14551 2143 1049

4957 2333 53 2447 331

139 1129 5717 4481

5791 149 727 1429

1289 14251 3947 1321

1097 353 643 59

1457181 1570143 1625615 1811933 1850825

359 2347 7561 24821 733

233 353 71 2341 827

239 109 439 19073 1489

353 2591 7193 8089

1885822 1920649 2134118

1471 2837 113

479 359 54721

2147188 2223285 2300608 2316257

23339 269 349 191

127 367 2300609 593

2507609 2575700 2683547 2721286 2774925

2879 599 4421 1373 4111 1321 379

11941 541 463 2131 487

65 66 67 68 69 70 71 72 73 74 75 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

675550 681088 722750 753505 766172 771457 867894 922129 942669 954087 993299 996091 1008988 1114271 1184610

2882422 2965675

7069 2551 769

3192189 3369359 3523174

5659 1063 15527 3001

1913 317 139 2659

3532407 3575518 3669327 3682461 3847270

197 193 3673 643 61

293 3575519

3053808 3058649

103 104

3063696 3112450

105 106 107 108 109 110

III

307 15131 3803 971 257

101 102

112

397 181

59 7109 3847271

1637 1283 53353 3767 563 347 137 14009 311 4603 1597 151 1447 223 9851 2351

1613 2269 1321

433

71 313 1913 1481 19699 73 2300611 647

811 829 281 1907 4447 181 15287 1021 1171

887 179 1217 14843 5437

139 7393 163 823

401 191 3559 7321

89 3575521

223

173 431 3847273

* *

Table 4. ctd

â&#x20AC;˘

Sen)

S(n+1)

S(n+2)

S(n+3)

113 114 115 116

3946899 3996604 3996924 4003753

131 13687

1361 2087

1327 2351

1181 15551 859 379

117 118 119

4083279 4089287 4176254 4231135

421 4089287 1087

149 271 1187 241 2003 1453

311 223 617

4319374 4330089 4407890

4243 3229 241 1021 773

69ll 761 3701 2549

2017 109 619 2659 1987 431

359 631 4964381 641 2003 1321

1229 439 ll549 2347 1373 277 24763 239

3739 20029 6703 8747 73

1741 9419 43 2731

25903 12671 797 7450169 6301

ll243 661 2729 13291

179 7450171 7643

3121 7717007 1249

1867 149 26953

1433 7717009 248473

443 3929 1401ll 2069 8840171 6719 199 6203 4201 4877 17569

9833 5273 365507 2903 347 137

863 787 18ll 829 8840173 10463 197 547

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

4460749 4466394 4497910 4527424 4964380 5041464 5223823 5225875 5567370 5808409 6086323 6149140 6278729 6598417 6611721 6662125 7019712 7083088 7208864 7450168 7535995 7699506 7717006 7951133 8161388 8246970 8406659

151 152 153 154 155 156 157

8822215 8840170 9050492 9369317 9558822

158

9944053

9616088 9739368

22871

14633 179 151 274177 10253 2131 227207 1663 349 3881 9369317 61 2027 109 2917

2063

2389 2333 45631

2243 199 4089289 79

1409 239 3011

13693 107

10631 ll047

3313 3761 211

677 181 3359

1223 349

1613 2027

241 617

499 4964383

239 109 433 3877

3061 3881 1319 1091 7297 6659 6143 479 919 26237 37853 20347 10847

13171 ll593 4951 967 1747 859 8017 7297

9369319 5717 107 4253 20089

6121 733 397

* *

Table 5. Smarandache function: 3-3 additive sextets for n<107

â&#x20AC;˘ 1 2 3 4 5 6 7 8 9 10 11

12 13 14

15 16 l7

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

Sen)

S (n+l)

S (n+2)

S (n+3)

S(n+4)

S(n+5)

5 5182 9855 10428 28373

5 2591 73

3 73

4 61

6 2593

5 19

53 61 3547 97

9859 163 1051 2963

683 461 727

809 337 4931

29 10433 2027 379 31

9161 3301

32589 83323 106488 113409 146572 257474 294742 448137 453250 465447 831096 1164960 1279039 1348296 1428620 1544770 1680357 1917568 2466880 2475373 3199719 3618482 4217047 4239054

859

11 10429 4729 3259 563

29 12601

1283 1031

36643 347

20939

463 127 479

3433 1223 224069 14621 797

1091 12281 448139 353 6947

4643 1021 571 2447 2393 1877 840179 1917569 2466881

21871 1669 691 499 2389 193 1680359 1559 4153

3041 479

41 2297 419 499 863

79 1669 71

569 101 37 1373 97 809 1279039 56179 1171 863 71 211 593 6173 15919 1973

2333 277

557 191

11941

17939 131 2221 421 7019 29 2243 82469 3301

1483 10739 971 108821 9439 1091 1999 45757 2087

8741057 8823577

809 1321 180073

35801 653

9171411 9975698

2179 947

5022920 5154719 5488091 6093975 6597860 6667100 6964515 7092334 7394240 7912020

9 9857

259517 1999 173

149 227 109 101

263 233 97 1613 107 617 673 347 49937 1607 1021 67 1917571 2466883 1181 13007 311 193 4993 1229

613 113 1307

3109 1987

271 11657 11149 1597 1063 883 15761

92333 23819 659 181 3061 509 16381

9967 23159 479 14251

547 441179 1277 3677

8669 448141 13331 59 8311 1283 1279043 139 3307 1433 1680361 1049 4637 6857 5087 593 317 3359 18199 4813 2423 7351 53 1877 4549 1801 139 7219 6607 257 577 523

977 8101 48859 1307 5171 224071 67 8951 17683 1543 911 9049 1039 479 840181 719 107 1217 601 3821 823 4643 607 3061 89 9829 4243 9733 1109 124427 5623 28771 4787 21313 2803 11171

Table 6. Smarandache fimction: 4-4 additive octets for n<10 7

1 2

Sen)

S(n+l)

S(n+2)

S(n+3)

S(n+4)

13 9 643 23 43 19 647 1487 347 359 137 89 673 1033 2221 67 167 701 54443 16333 23333 349 279577 10753 2273 1997 3539 323294 1483 3079 10103 1913 5987 368680 709 2903 1699 1429 1511 857434 8089 769 71453 11587 353 1545493 1545493 1669 359 3167 389 2458284 204857 28921 53441 467 21011 3546232 19273 3546233 3863 151 1609 3883322 8707 3709 12289 155333 2287 4945200 3299 673 317 9851 139 5219814 2411 5483 4339 1259 2003 6055151 128833 1249 465781 432511 14951 6572015 3137 461 31147 6277 523 7096751 7096751 223 457 506911 473117 7217695 4021 4799 2131 3608849 191 7530545 383 175129 6947 547 5953 23 643 10409 44418 163329 279577

3 4 5 6 7 8 9 10 11 12 13

14 15 16 17 18 19 20

S(n+7)

S(n+5)

S (n+6)

9 127 1433

59 2083 617 10889

81667 2741 10429 2731

279583 61 2221

2887

233

4519

1545499

31 1777 1201 8737 101 277 65957

281 223481 2339 81943 16649 5023 3546239 143827 32633 1291 5717 6197 1019 241 617 10631 1559 2671 1009193 157 24251 4583 4397 30071 7096757 10267 3608851 491 2551 150611 34703

Table 7. Smarandache fimction: 5-5 additive relationships for n<l 0 7 I

S(n+l.)

S(n+l)

SIn)

S (n+l)

S(n+2)

S(n+3)

S(n+4)

S(n+5)

S (n+6)

S(n+7)

570

571

191

577

193

1230

1231

137

617

103

1237

619

392152

49019

392153

9337

733

43573

15083

392159

463

1984525

487

992263

2371

1091

797

701

2441

992267

4730276

5303

54371

17783

36109

39419

3011

2819

6653

5351

135151

5798379

8087

499

2339

2677

2417

8839

139

587

2927

3527

5838665

7253

7103

227

132697

107

4457

9463

17377

37189

78901

Table 8. Smarandache fimction: 6-6 additive relationships for n<107

â&#x20AC;˘

SIn)

158

163

167

20873

167

307

6959

20879

157

197

6961

227

21633

7211

373

4327

601

281

349

7213

541

3607

941

773

103515

103

3697

1697

7963

647

3137

271

643

8627

101

1399

132899

10223

443

383

863

14767

449

1399

1303

4583

223

6329

13291

3681 ?7

661

61363

353

449

3719

9689

1301

73637

107

1109

1447

593

15107

3253

643

3323

1193

10837

293

5559383

5387

5559373 5559373

S(n+1) 8 (n+2) S(n+3)

S(n+4) S(n+5) S(n+6) S(n+7) S(n+8) S(0+9) S(0+10)

5(0+11)

Table 9. Smarandache function 50-50 additive relations

Sum

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 29 383 479 71 137 101 B

113 62 641 37 11 20307

n~58631

n-16539

n 1876 5(n)/5(n+51) 5(n+l)/5(n+52) 5(n+2) 15 (n+53) S(n+3)/5(n+54) 5 (n+4) /5 (n+55) 5 (n+5) /5 (n+56) S(n+6)/S(n+57) S(n+7)/S(n+58) 5 (n+8) /5 (n+59) S(n+9)/S(n+60) S(n+10)/5(n+61) 5 (n+11) /5 (n+62) S(n+12)/S(n+63) S(n+13)/S(n+64) 5 (n+14) /5 (n+65) S(n+15)/S(n+66) S(n+16)/S(n+67) S(n+17/S(n+68) 5 (n+18) /5 (n+69) 5(n+19)/S(n+70) 5 (n+20)/5(n+71) S(n+21)/S(n+72) S(n+22)/S(n+73) S(n+23)/S(n+74) S(n+24)/S(n+75) 5 (n+25) /5 (n+76) S(n+26)/S(n+77) S(n+27)/S(n+78) S(n+28)/S(n+79) S(n+29)/S(n+80) S(n+30)/S(n+81) S(n+31)/S(n+82) S(n+32)/S(n+83) S(n+33)/S(n+84) S(n+34)/S(n+85) S(n+35)/S(n+86) S(n+36)/S(n+87) S(n+37)/S(n+88) S(n+38)/5(n+89) 5(n+39)/S(n+90) 5 (n+40) /S (n+91) S(n+41)/S(n+92) S(n+42)/S(n+93) S(n+43)/S(n+94) S(n+44)/S(n+95) 5(n+45)/S(n+96) 5 (n+46) /S (n+97) 5(n+47)/5(n+98) 5(n+48)/5(n+99) 5 (n+49) /5 (n+100)

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 1951 61 31 977 23 163 103 89 653 14 53 109 151 491 131 983 281 41 179 197 73 29 1973 47 79 20307

149 827 139 919 233 47 1103 8273 16547 197 67 331 613 2069 16553 89 43 4139 5519 487 571 23 16561 26 5521 101 3313 251 16567 109 263 1657 227 1381 16573 8287 17

37 137 307 281 829 5527 8291 103 691 107 8293 97 29 154521

313 79 353 61 5531 8297 3319 461 2371 193 503 83 1277 2767 16603 593 41 38 16607 173 977 151 113 4153 449 71 3323 67 191 1187 16619 277 1511 8311 1847 1039 19 163 1279 4157 241 1663 16631 11 16633 8317 1109 4159 131 59 154521

58631 349 3449 1543 1303 137 307 337 8377 733 1777 269 347 181 317 71 173 7331 263 23 659 43 21 29327 11731 47 58657 211 19553 419 58661 3259 5333 7333 3911 29333 34 4889 4513 5867 53 193 2551 127 2347 14669 19559 29339 58679 163 457399

101 61 631 863 97 9781 58687 131 6521 5869 3089 73 58693 29347 43 29 743 1087 58699 587 1151 599 1249 1223 199 197 593 1129 8387 103 58711 179 19571 947 11743 233 827 157 46 367 4517 9787 8389 277 29 29363 58727 2447 281 839 457399

n-l09606 7829 1523 2549 2069 4567 54829 109609 3323 5483 113 641 109661 373 409 109663 2237 149 18269 2437 1993 31 54833 6451 599 2383 37 109619 15667 997 29 109621 263 929 13709 36541 109673 193 677 107 877 151 3917 36559 15661 61 27407 1637 937 577 457 59 2963 317 571 6449 1741 1613 191 7309 21937 27409 181 9967 251 6091 13711 109639 36563 2741 1567 479 227 277 4217 1321 2551 4219 9137 21929 103 751 857 131 15671 89 389 673 199 1097 43 2333 239 347 54851 36551 9973 503 653 593 241 705120 705120

n~2385965

1087 36151 140351 11471 795323 601 1213 347 8431 51869 1097 298247 6997 56809 2385979 119299 20393 1697 2385983 43 68171 2657 795329 257 2385989 8837 2385991 311 7433 563 113 198833 457 2693 313 1193 89 8461 2386003 307 2179 251 3463 1069 2386009 199 795337 596503 340859 727 18703984

References I. . H. Ibstedt, The Smarandache FWlction S(n), Smarandache Function Journal, Vol. 2-3, No 1, pgs 43-50. 2. M. Bencze, Smarandache Relationships and Subsequences, Smarandache Notions Journal, Vol. 11, No 1-2-3, pgs 79-85. 3. H. Ibstedt, Non-Recursive Sequences, Computer Analysis of Number Sequences, American Research Press, 1998.

7823 431 1091 70177 1093 23 216911 1193011 1151 2953 95441 1867 7927 596507 795343 887 2386031 4519 6329 1301 3119 14549 30203 397673 4051 59651 113621 1193021 9431 22093 477209 91771 795349 2663 50767 15907 2386051 35089 265117 108457 9739 2687 2386057 62791 317 2251 2386061 653 2386063 757 18703984