Bernoulli numbers and arithmetic progressions
cino hilliard
hillcino368 at hotmail.com
Thu Feb 5 10:48:03 CET 2004
Hi,
>From: Roland Bacher <Roland.Bacher at ujf-grenoble.fr>
>To: seqfan at ext.jussieu.fr
>Subject: Bernoulli numbers and arithmetic progressions
>Date: Wed, 4 Feb 2004 12:55:29 +0100
>
>
>Here an amusing remark using the data below:
>
>all values yielding 37 are of the form: 574+666*k, k=0,1,2,3,4,...
>and form thus an arithmetic progression with step 666=18*37=((37-1)/2)*37;
>
>all values yielding 59 are of the form: 1269+1711*k, k=0,1,2,3
>and 1711=28*59=((59-1)/2)*59
>
>the two values yielding 67 are at distance 2211=((67-1)/2)*67
>
>Conjecture: all indices yielding a given prime p form an arithmetic
>progression of step ((p-1)/2)*p.
>
>Is this obvious?
This does not compute 8557 and 157 from first born 4789 and 157
>
>If yes, the only important data would be the beginning of the
>arithmetic progression for a given prime.
>
>By the way, in order to check this conjecture experimentally in
>a few more cases it is enough to compute the corresponding
>p-valuations. This can be done using (for instance) a recursive
>formula for Bernoulli numbers and working in the finite ring
>Z/(p^a Z) with a>0 big enough (in any case, a such that p^a>2n
>should work). This trick should allow to push computations quite a lot
>further.
>
I computed some more terms up to 11000 trials and one term past 20000 for
20090,59.
The index = (p-1)/2*k + index first born, k=0,1,2,.. works in all cases but
does not
necessarily catch all indices of the same irreg prime.The first miss in 1 to
10000 is first born
4789 and 157. This does not catch 8557 and 157.Could it be that the first
born cannot be
prime and a composite first born must be found? I included 8557 and 157 in
the spreadsheet.
From 10000 to 11000 10587,683 occurs but is not forcasted since it _is_
first born and had no
calculation antecedent. My guess is this "missing" will prevall and will
not be able to predict all
future terms with certainty.
The spread sheet can be found at
http://groups.yahoo.com/group/B2LCC/files/Bernoulli/
Excel calc Excel calc
Excel calc
Pari calc Pari calc (p-1)/2*p*k sorted Factors
of
(n) Index Irr Prime Irr Prime Index Index
---- -------- ------------ ------------- -----------
--------------------
1 574 37 37 574 2, 7, 41
2 1185 103 103 1185 3,5, 79
3 1240 37 37 1240 2,2,2,5, 31
4 1269 59 59 1269 3,3,3, 47
5 1376 131 131 1376 2,2,2,2,2, 43
6 1906 37 37 1906 2, 953
7 1910 67 67 1910 2,5, 191
8 2572 37 37 2572 2,2, 643
9 2689 283 283 2689 2689
10 2980 59 59 2980 2,2,5, 149
11 3238 37 37 3238 2, 1619
12 3384 101 101 3384 2,2,2,3,3, 47
13 3801 691 691 3801 3, 7, 181
14 3904 37 37 3904 2,2,2,2,2,2, 61
15 4121 67 67 4121 13, 317
16 4570 37 37 4570 2,5, 457
17 4691 59 59 4691 4691
18 4789 157 157 4789 4789
19 5236 37 37 5236 2,2, 7, 11, 17
20 5862 617 617 5862 2,3, 977
21 5902 37 37 5902 2, 13, 227
22 6227 593 593 6227 13, 479
23 6332 67 67 6332 2,2, 1583
24 6402 59 59 6402 2,3, 11, 97
25 6438 103 103 6438 2,3, 29, 37
26 6568 37 37 6568 2,2,2, 821
27 7234 37 37 7234 2, 3617
28 7900 37 37 7900 2,2,5,5, 79
29 8113 59 59 8113 7, 19, 61
30 8434 101 101 8434 2, 4217
31 8543 67 67 8543 8543
32 8557 157 157 8557 43, 199 *Failed the test
33 8566 37 37 8566 2, 4283
34 9232 37 37 9232 2,2,2,2, 577
35 9611 149 149 9611 7, 1373
36 9670 233 233 9670 2,5, 967
37 9824 59 59 9824 2,2,2,2,2, 307
38 9891 131 131 9891 3,3, 7, 157
39 9898 37 37 9898 2, 7, 7, 101
40 10564 37 37 10564 2,2, 19, 139
41 10587 683 67 10587 3,3529 *failed the test
42 10754 67 67 10754 2, 19, 283
37 11230 2,5, 1123
59 11535 3,5, 769
103 11691 3,3,3, 433
37 11896 2,2,2, 1487
37 12562 2, 11, 571
67 12965 5, 2593
37 13228 2,2, 3307
59 13246 2, 37, 179
101 13484 2,2, 3371
37 13894 2, 6947
37 14560 2,2,2,2,2,5, 7, 13
59 14957 14957
67 15176 2,2,2, 7, 271
37 15226 2, 23, 331
37 15892 2,2, 29, 137
37 16558 2, 17, 487
59 16668 2,2,3,3, 463
103 16944 2,2,2,2,3, 353
157 17035 5, 3407
37 17224 2,2,2, 2153
67 17387 17387
37 17890 2,5, 1789
59 18379 18379
131 18,406 2, 9203
101 18534 2,3, 3089
37 18556 2,2, 4639
37 19222 2, 7, 1373
67 19598 2, 41, 239
37 19888 2,2,2,2, 11, 113
59 20090 2,5, 7, 7, 41
37 20554 2, 43, 239
149 20637 3,3, 2293
157 20803 71, 293
37 21220 2,2,5, 1061
59 21801 3, 13, 13, 43
67 21809 113, 193
37 21886 2, 31, 353
103 22197 3, 7, 7, 151
37 22552 2,2,2, 2819
37 23218 2, 13, 19, 47
59 23512 2,2,2, 2939
101 23584 2,2,2,2,2, 11, 67
37 23884 2,2, 7, 853
67 24020 2,2,5, 1201
37 24550 2,5,5, 491
37 25216 2,2,2,2,2,2,2, 197
59 25223 11, 2293
37 25882 2, 12941
67 26231 17, 1543
37 26548 2,2, 6637
131 26921 26921
59 26934 2,3, 67, 67
37 27214 2, 11, 1237
103 27450 2,3,3,5,5, 61
37 27880 2,2,2,5, 17, 41
67 28442 2, 14221
37 28546 2, 7, 2039
101 28634 2, 103, 139
59 28645 5, 17, 337
37 29212 2,2, 67, 109
157 29281 7, 47, 89
37 29878 2, 14939
59 30356 2,2, 7589
37 30544 2,2,2,2, 23, 83
67 30653 7, 29, 151
37 31210 2,5, 3121
149 31663 31663
59 32067 3,3, 7, 509
103 32703 3, 11, 991
67 32864 2,2,2,2,2, 13, 79
157 33049 33049
101 33684 2,2,3, 7, 401
59 33778 2, 16889
67 35075 5,5, 23, 61
131 35436 2,2,3, 2953
59 35489 23, 1543
233 36698 2, 59, 311
59 37200 2,2,2,2,3,5,5, 31
67 37286 2, 103, 181
103 37956 2,2,3, 3163
101 38734 2, 107, 181
59 38911 167, 233
67 39497 127, 311
59 40622 2, 19, 1069
157 41527 131, 317
67 41708 2,2, 10427
59 42333 3, 103, 137
283 42592 2,2,2,2,2, 11, 11, 11
149 42689 42689
103 43209 3,3, 4801
101 43784 2,2,2, 13, 421
67 43919 37, 1187
131 43951 43951
59 44044 2,2, 7, 11, 11, 13
157 45295 5, 9059
59 45755 5, 9151
67 46130 2,5, 7, 659
59 47466 2,3,3,3,3, 293
67 48341 48341
103 48462 2,3, 41, 197
101 48834 2,3,3, 2713
59 49177 49177
67 50552 2,2,2, 71, 89
59 50888 2,2,2, 6361
131 52466 2, 37, 709
59 52599 3, 89, 197
67 52763 19, 2777
149 53715 3,5, 3581
103 53715 3,5, 3581
157 53773 53773
101 53884 2,2, 19, 709
59 54310 2,5, 5431
67 54974 2, 27487
59 56021 7, 53, 151
67 57185 5, 11437
157 57541 11, 5231
59 57732 2,2,3, 17, 283
101 58934 2, 79, 373
103 58968 2,2,2,3,3,3,3, 7, 13
67 59396 2,2, 31, 479
59 59443 59443
131 60981 3, 20327
59 61154 2, 30577
67 61607 7, 13, 677
59 62865 3,3,5, 11, 127
233 63726 2,3, 13, 19, 43
67 63818 2, 17, 1877
101 63984 2,2,2,2,3, 31, 43
103 64221 3, 21407
59 64576 2,2,2,2,2,2, 1009
149 64741 101, 641
157 66019 107, 617
67 66029 66029
59 66287 13, 5099
59 67998 2,3, 7, 1619
67 68240 2,2,2,2,5, 853
101 69034 2, 7, 4931
103 69474 2,3, 11579
131 69496 2,2,2, 7, 17, 73
59 69709 69709
157 69787 19, 3673
67 70451 70451
59 71420 2,2,5, 3571
67 72662 2, 47, 773
59 73131 3, 19, 1283
101 74084 2,2, 18521
103 74727 3,3, 19, 19, 23
59 74842 2, 23, 1627
67 74873 74873
149 75767 75767
59 76553 37, 2069
67 77084 2,2, 7, 2753
131 78011 181, 431
59 78264 2,2,2,3,3, 1087
157 78265 5, 11, 1423
101 79134 2,3, 11, 11, 109
67 79295 5, 15859
59 79975 5,5, 7, 457
103 79980 2,2,3,5, 31, 43
67 81506 2, 83, 491
157 82033 7, 11719
283 82495 5, 7, 2357
67 83717 83717
101 84184 2,2,2, 17, 619
103 85233 3, 28411
67 85928 2,2,2, 23, 467
131 86526 2,3,3, 11, 19, 23
149 86793 3, 7, 4133
67 88139 53, 1663
101 89234 2, 44617
67 90350 2,5,5, 13, 139
103 90486 2,3,3, 11, 457
157 90511 90511
233 90754 2, 45377
67 92561 7, 7, 1889
157 94279 29, 3251
101 94284 2,2,3,3,3,3,3, 97
67 94772 2,2, 19, 29, 43
131 95041 101, 941
103 95739 3, 7, 47, 97
67 96983 293, 331
149 97819 23, 4253
67 99194 2, 49597
101 99334 2, 49667
103 100992 2,2,2,2,2,2,2,3, 263
67 101405 5, 17, 1193
157 102757 211, 487
131 103556 2,2, 25889
67 103616 2,2,2,2,2,2, 1619
101 104384 2,2,2,2,2,2, 7, 233
103 106245 3,3,3,5, 787
157 106525 5,5, 4261
149 108845 5, 11, 1979
101 109434 2,3, 13, 23, 61
103 111498 2,3, 18583
131 112071 3, 37357
101 114484 2,2, 28621
157 115003 7, 7, 2347
103 116751 3, 38917
233 117782 2, 7, 47, 179
157 118771 73, 1627
101 119534 2, 59, 1013
149 119871 3,3, 19, 701
131 120586 2, 60293
103 122004 2,2,3,3, 3389
283 122398 2, 19, 3221
101 124584 2,2,2,3, 29, 179
157 127249 127249
103 127257 3, 13, 13, 251
131 129101 7, 18443
101 129634 2, 64817
149 130897 13, 10069
157 131017 37, 3541
103 132510 2,3,5, 7, 631
101 134684 2,2, 11, 3061
131 137616 2,2,2,2,3, 47, 61
103 137763 3,3, 15307
157 139495 5, 23, 1213
101 139734 2,3,3, 7, 1109
149 141923 347, 409
103 143016 2,2,2,3, 59, 101
157 143263 143263
101 144784 2,2,2,2, 9049
233 144810 2,3,3,5, 1609
131 146131 29, 5039
103 148269 3, 11, 4493
101 149834 2, 19, 3943
157 151741 41, 3701
149 152949 3, 17, 2999
103 153522 2,3,3,3, 2843
131 154646 2, 77323
101 154884 2,2,3, 12907
157 155509 155509
103 158775 3,5,5, 29, 73
101 159934 2, 79967
283 162301 109, 1489
131 163161 3,3,3, 6043
149 163975 5,5, 7, 937
157 163987 163987
103 164028 2,2,3, 13669
101 164984 2,2,2, 41, 503
157 167755 5, 7, 4793
103 169281 3,3, 7, 2687
101 170034 2,3, 17, 1667
131 171676 2,2, 167, 257
233 171838 2, 151, 569
103 174534 2,3, 19, 1531
149 175001 139, 1259
101 175084 2,2, 7, 13, 13, 37
157 176233 29, 59, 103
103 179787 3, 59929
157 180001 180001
101 180134 2, 90067
131 180191 11, 16381
593 181755 3,3,5, 7, 577
103 185040 2,2,2,2,3,3,5, 257
101 185184 2,2,2,2,2,3,3, 643
149 186027 3, 59, 1051
157 188479 17, 11087
131 188706 2,3, 7, 4493
101 190234 2, 11, 8647
103 190293 3, 137, 463
157 192247 11, 17477
101 195284 2,2, 48821
103 195546 2,3, 13, 23, 109
617 195898 2, 41, 2389
149 197053 193, 1021
131 197221 197221
233 198866 2, 17, 5849
101 200334 2,3, 173, 193
157 200725 5,5, 7, 31, 37
103 200799 3,3,3,3, 37, 67
283 202204 2,2, 50551
157 204493 17, 23, 523
101 205384 2,2,2, 25673
131 205736 2,2,2, 25717
103 206052 2,2,3, 7, 11, 223
149 208079 251, 829
101 210434 2, 7, 15031
103 211305 3,5, 14087
157 212971 11, 19, 1019
131 214251 3, 17, 4201
101 215484 2,2,3, 17957
103 216558 2,3,3, 53, 227
157 216739 193, 1123
149 219105 3,3,3,3,5, 541
101 220534 2, 31, 3557
103 221811 3, 107, 691
131 222766 2, 31, 3593
157 225217 225217
101 225584 2,2,2,2, 23, 613
233 225894 2,3, 37649
103 227064 2,2,2,3, 9461
157 228985 5, 41, 1117
149 230131 11, 20921
101 230634 2,3,3,3, 4271
131 231281 41, 5641
103 232317 3,3, 83, 311
101 235684 2,2, 58921
157 237463 83, 2861
103 237570 2,3,5, 7919
131 239796 2,2,3,3, 6661
149 241157 7, 47, 733
157 241231 149, 1619
283 242107 239, 1013
691 242196 2,2,3, 20183
103 242823 3, 7, 31, 373
131 248311 7, 19, 1867
157 249709 67, 3727
149 252183 3, 84061
233 252922 2, 126461
157 253477 7, 7, 7, 739
131 256826 2, 128413
157 261955 5, 52391
149 263209 263209
131 265341 3, 241, 367
157 265723 467, 569
131 273856 2,2,2,2,2,2, 11, 389
157 274201 274201
149 274235 5, 13, 4219
157 277969 233, 1193
233 279950 2,5,5, 11, 509
283 282010 2,5, 28201
131 282371 23, 12277
149 285261 3, 95087
157 286447 7, 151, 271
157 290215 5, 58043
131 290886 2,3, 48481
149 296287 296287
157 298693 298693
131 299401 299401
157 302461 19, 15919
233 306978 2,3, 7, 7309
149 307313 29, 10597
131 307916 2,2, 7, 7, 1571
157 310939 239, 1301
157 314707 314707
131 316431 3,3, 35159
149 318339 3,3, 7, 31, 163
283 321913 89, 3617
157 323185 5, 109, 593
131 324946 2, 162473
157 326953 11, 29723
149 329365 5, 19, 3467
131 333461 139, 2399
233 334006 2, 23, 53, 137
157 335431 509, 659
157 339199 7, 47, 1031
149 340391 17, 20023
131 341976 2,2,2,3, 14249
157 347677 11, 31607
131 350491 463, 757
149 351417 3, 11, 23, 463
157 351445 5, 70289
593 357283 357283
131 359006 2, 17, 10559
157 359923 53, 6791
233 361034 2, 97, 1861
283 361816 2,2,2, 7, 7, 13, 71
149 362443 362443
157 363691 363691
131 367521 3, 7, 11, 37, 43
157 372169 7, 79, 673
149 373469 41, 9109
157 375937 31, 67, 181
131 376036 2,2, 94009
157 384415 5, 76883
149 384495 3,5, 25633
131 384551 239, 1609
617 385934 2, 17, 11351
233 388062 2,3,3, 21559
157 388183 388183
131 393066 2,3,3,3, 29, 251
149 395521 7, 56503
157 396661 17, 23333
157 400429 400429
283 401719 73, 5503
149 406547 406547
157 408907 461, 887
157 412675 5,5, 17, 971
233 415090 2,5, 13, 31, 103
149 417573 3,3, 13, 43, 83
157 421153 23, 18311
157 424921 7, 60703
149 428599 67, 6397
157 433399 433399
157 437167 137, 3191
149 439625 5,5,5, 3517
283 441622 2, 220811
233 442118 2, 221059
157 445645 5, 19, 4691
157 449413 29, 15497
149 450651 3, 150217
157 457891 7, 65413
157 461659 11, 41969
149 461677 461677
233 469146 2,3, 78191
157 470137 229, 2053
149 472703 7, 7, 11, 877
157 473905 5, 94781
691 480591 3,3, 67, 797
283 481525 5,5, 11, 17, 103
157 482383 11, 43853
149 483729 3, 383, 421
157 486151 23, 23, 919
157 494629 43, 11503
149 494755 5, 53, 1867
233 496174 2, 7, 7, 61, 83
157 498397 498397
149 505781 505781
157 506875 5,5,5,5, 811
157 510643 7, 72949
149 516807 3,3,3, 19141
157 519121 519121
283 521428 2,2, 61, 2137
157 522889 613, 853
233 523202 2, 261601
157 531367 29, 73, 251
593 532811 532811
157 535135 5, 19, 43, 131
157 543613 7, 77659
157 547381 307, 1783
233 550230 2,3,5, 18341
157 555859 71, 7829
157 559627 53, 10559
283 561331 41, 13691
157 568105 5, 113621
157 571873 571873
617 575970 2,3,5, 73, 263
233 577258 2, 11, 19, 1381
283 601234 2, 521, 577
233 604286 2, 302143
233 631314 2,3,3,3,3,3,3, 433
283 641137 7, 91591
233 658342 2, 17, 17, 17, 67
283 681040 2,2,2,2,5, 8513
233 685370 2,5, 7, 9791
593 708339 3, 17, 17, 19, 43
233 712398 2,3, 37, 3209
691 718986 2,3, 119831
283 720943 720943
233 739426 2, 457, 809
283 760846 2, 380423
617 766006 2, 29, 47, 281
233 766454 2, 13, 41, 719
233 793482 2,3, 132247
283 800749 397, 2017
233 820510 2,5, 82051
283 840652 2,2, 29, 7247
233 847538 2, 423769
233 874566 2,3,3, 7, 11, 631
283 880555 5, 13, 19, 23, 31
593 883867 23, 83, 463
233 901594 2, 450797
283 920458 2, 7, 11, 43, 139
233 928622 2, 464311
233 955650 2,3,5,5, 23, 277
617 956042 2, 19, 139, 181
691 957381 3, 319127
283 960361 131, 7331
233 982678 2, 491339
283 1000264 2,2,2, 97, 1289
233 1009706 2, 504853
233 1036734 2,3, 131, 1319
283 1040167 1040167
593 1059395 5, 211879
233 1063762 2, 7, 75983
283 1080070 2,5, 108007
233 1090790 2,5, 19, 5741
233 1117818 2,3,3, 13, 17, 281
283 1119973 163, 6871
233 1144846 2, 572423
617 1146078 2,3,3, 63671
283 1159876 2,2, 17, 37, 461
233 1171874 2, 11, 53267
691 1195776 2,2,2,2,2,2,2,2,3,3,3, 173
233 1198902 2,3, 211, 947
283 1199779 7, 101, 1697
233 1225930 2,5, 43, 2851
593 1234923 3, 411641
283 1239682 2, 619841
233 1252958 2, 7, 31, 2887
283 1279585 5, 255917
283 1319488 2,2,2,2,2,2, 53, 389
617 1336114 2, 13, 13, 59, 67
283 1359391 11, 123581
283 1399294 2, 13, 53819
593 1410451 7, 201493
691 1434171 3, 17, 61, 461
283 1439197 607, 2371
283 1479100 2,2,5,5, 7, 2113
283 1519003 811, 1873
617 1526150 2,5,5, 131, 233
283 1558906 2, 83, 9391
593 1585979 59, 26881
283 1598809 929, 1721
283 1638712 2,2,2, 19, 10781
691 1672566 2,3, 7, 7, 5689
283 1678615 5, 113, 2971
617 1716186 2,3, 103, 2777
283 1718518 2, 859259
283 1758421 7, 251203
593 1761507 3,3,3,3,3, 11, 659
283 1798324 2,2, 11, 23, 1777
283 1838227 17, 108131
617 1906222 2, 953111
691 1910961 3,3, 13, 16333
593 1937035 5, 31, 12497
617 2096258 2, 1048129
593 2112563 29, 97, 751
691 2149356 2,2,3, 11, 19, 857
617 2286294 2,3, 463, 823
593 2288091 3, 13, 13, 4513
691 2387751 3, 795917
593 2463619 2463619
617 2476330 2,5, 247633
691 2626146 2,3,3, 145897
593 2639147 7, 377021
617 2666366 2, 971, 1373
593 2814675 3,5,5, 37529
617 2856402 2,3,3, 31, 5119
691 2864541 3, 954847
593 2990203 829, 3607
617 3046438 2, 1523219
691 3102936 2,2,2,3, 129289
593 3165731 263, 12037
617 3236474 2, 457, 3541
593 3341259 3,3, 371251
691 3341331 3,3,3,3, 7, 71, 83
617 3426510 2,3,5, 114217
593 3516787 3516787
691 3579726 2,3, 53, 11257
617 3616546 2, 17, 17, 6257
593 3692315 5, 11, 11, 17, 359
617 3806582 2, 13, 146407
691 3818121 3, 167, 7621
593 3867843 3, 7, 67, 2749
617 3996618 2,3, 23, 28961
593 4043371 19, 139, 1531
691 4056516 2,2,3,3, 281, 401
617 4186654 2, 2093327
593 4218899 4218899
691 4294911 3, 1431637
617 4376690 2,5, 541, 809
593 4394427 3, 1464809
691 4533306 2,3, 755551
617 4566726 2,3,3,3, 19, 4451
593 4569955 5, 13, 167, 421
593 4745483 1019, 4657
617 4756762 2, 197, 12073
691 4771701 3,3, 11, 157, 307
593 4921011 3,3, 23, 23773
617 4946798 2, 89, 27791
691 5010096 2,2,2,2,3, 7, 13, 31, 37
593 5096539 7, 7, 47, 2213
617 5136834 2,3, 856139
691 5248491 3, 1749497
593 5272067 41, 149, 863
617 5326870 2,5, 532687
593 5447595 3,5, 363173
691 5486886 2,3,3,3, 17, 43, 139
617 5516906 2, 2758453
593 5623123 11, 511193
617 5706942 2,3, 257, 3701
691 5725281 3, 41, 89, 523
593 5798651 107, 54193
617 5896978 2, 2948489
691 5963676 2,2,3, 29, 17137
593 5974179 3, 199, 10007
617 6087014 2, 3043507
593 6149707 6149707
691 6202071 3,3, 499, 1381
617 6277050 2,3,3,5,5, 13, 29, 37
593 6325235 5, 7, 127, 1423
691 6440466 2,3, 1073411
617 6467086 2, 173, 18691
593 6500763 3,3,3, 240769
617 6657122 2, 3328561
593 6676291 17, 392723
691 6678861 3, 7, 19, 19, 881
617 6847158 2,3, 17, 67129
593 6851819 13, 527063
691 6917256 2,2,2,3,3, 191, 503
593 7027347 3, 643, 3643
617 7037194 2, 3518597
691 7155651 3, 257, 9281
593 7202875 5,5,5, 29, 1987
617 7227230 2,5, 722723
593 7378403 19, 31, 12527
691 7394046 2,3, 11, 112031
617 7417266 2,3, 1236211
593 7553931 3, 7, 11, 53, 617
617 7607302 2, 43, 53, 1669
691 7632441 3,3,3, 282683
593 7729459 73, 105883
617 7797338 2, 151, 25819
691 7870836 2,2,3, 59, 11117
593 7904987 7904987
617 7987374 2,3,3, 41, 79, 137
593 8080515 3,3,5, 79, 2273
691 8109231 3, 13, 337, 617
617 8177410 2,5, 19, 193, 223
691 8347626 2,3,3, 7, 97, 683
617 8367446 2, 23, 101, 1801
617 8557482 2,3, 1426247
691 8586021 3, 1193, 2399
617 8747518 2, 13, 31, 10853
691 8824416 2,2,2,2,2,3, 91921
691 9062811 3,3, 1006979
691 9301206 2,3, 47, 32983
691 9539601 3, 17, 17, 11003
691 9777996 2,2,3,3,3,3, 103, 293
691 10016391 3, 7, 11, 131, 331
691 10254786 2,3, 1709131
691 10493181 3,3, 1165909
691 10731576 2,2,2,3, 149, 3001
691 10969971 3, 3656657
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