Bernoulli numbers and arithmetic progressions

[T. D. Noe]

>i believe (and have suggested to the list)
>that Kummer's Congruence explains the answer
>
>the simplest way to see the question
>is to look at sequences A090496 A090495 which are
>based on the surprising fact that A001067 and A046968
>agree for the first 574 or so terms but then differ
>
>it is also clear i think that the terms of A090496
>are products of irregular primes, although so far only single
>primes have shown up


For irregular prime p, I think that p first appears as a quotient in
A090496 for

n = p*k - (p^2-1)/2,

where k is least integer such that k > (p+1)/2 and p | numerator(B(2k)).
Note that

2n-1 = p*(2k-p)

For the first 65 irregular primes, we have

p	k	n
37	34	574
59	51	1269
67	62	1910
101	84	3384
103	63	1185
131	76	1376
149	139	9611
157	109	4789
233	158	9670
257	210	20946
263	181	13019
271	177	11247
283	151	2689
293	224	22708
307	197	13355
311	301	45251
347	313	48407
353	269	32653
379	239	18761
389	294	38706
401	391	76391
409	267	25563
421	330	50310
433	399	79023
461	328	44948
463	296	29864
467	280	21716
491	391	71441
523	461	104339
541	313	22993
547	408	73572
557	389	61549
577	314	14714
587	338	26122
593	307	6227
607	599	179369
613	567	159687
617	318	5862
619	523	132157
631	355	24925
647	441	76023
653	350	15346
659	441	73479
673	540	136956
677	652	212240
683	357	10587
691	351	3801
727	552	137040
751	520	108520
757	635	194171
761	510	98550
773	752	282532
797	508	87272
809	569	133081
811	677	220187
821	782	305002
827	464	41764
839	452	27268
877	872	380180
881	521	70921
887	652	184940
929	724	241076
953	554	73858
971	568	80108
1061	767	250927

I have verified the small values of n, but don't have the time/power to
check the larger values.  Here is the Mathematica code:

c={37, 59, 67, 101, 103, 131, 149, 157, 233, 257, 263, 271, 283, 293, 307,
311, 347, 353, 379, 389, 401, 409, 421, 433, 461, 463, 467, 491, 523, 541,
547, 557, 577, 587, 593, 607, 613, 617, 619, 631, 647, 653, 659, 673, 677,
683, 691, 727, 751, 757, 761, 773, 797, 809, 811, 821, 827, 839, 877, 881,
887, 929, 953, 971, 1061};
Do[p=c[[i]]; k=(p+3)/2;
  While[Mod[Numerator[BernoulliB[2k]], p] > 0, k++];
  Print[p, " ", k, " ", p*k - (p^2-1)/2], {i, Length[c]}]

Tony

Tony Noe                  | voice:     503-690-2099
Software Spectra, Inc.    | fax:       503-690-8159
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