Bernoulli numbers and arithmetic progressions

[Richard Guy]

Evidently yet another manifestation of
the Law of Small Numbers.  R.

On Sun, 8 Feb 2004, Hans Havermann wrote:

> I asked:
> 
> > What are the values of p*k-(p-1)/2 where 
> > Mod[Numerator[BernoulliB[2k]], p] == 0 for p=37?
> >
> > {574, 1240, 1351, 1906, 2572, 2720, 3238, 3904, 4089, 4570, 5236, 
> > 5458, 5902, 6568, 6827, 7234, 7900, 8196, 8566, 9232, 9565, 9898, ...}
> >
> > Every third term in this calculated sequence, it seems, is a number 
> > that does not figure in the *actual* 37-appearance sequence.
> 
> I was still convinced however, ignoring every third term, that 37 
> appears periodically with a difference of 666 between adjacent terms. 
> As I calculated and collated the differences between adjacent terms of 
> a range of irregular primes, hoping to differentiate the clearly 
> periodic ones from the aberrations, I ran into a rude surprise as I 
> upped my values of k. Same calculation as above, ignoring every third 
> term, the differences between adjacent terms for p=37 are:
> 
> {666, 666, 666, 666, 666, 666, 666, 666, 666, 666, 666, 666, 666, 666, 
> 666, 666, 666, 666, 666, 666, 666, 666, 666, 666, 666, 666, 666, 666, 
> 666, 666, 666, 1332, 666, 666, 666, 1332, 111, 1221, 148, 1184, 185, 
> 1147, 222, 1110, 259, 1073, 296, 1036, 333, 999, 370, 962, 407, ...}
> 
> I feel a little discombobulated and would appreciate someone confirming 
> or denying this progression.
>