Bernoulli numbers and arithmetic progressions

[N. J. A. Sloane]

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

Neil

JHC wrote:
    What is the conjecture being spoken of here?  It sounds as though it's
some kind of congruence involving Bernoulli numbers, in which case I'd 
like to have a shot at it.