Bernoulli numbers and arithmetic progressions

[T. D. Noe]

I think Hans is on the right track.  I'm fairly sure that some
computational errors obscure the truth.  It appears that the values of n in
A090495 that yield the same irregular prime quotient _do_ form arithmetic
progressions.  It appears that the irregularity index (see A073276 ,
A073277, A060975, A061576, and just-submitted A091887) of an irregular
prime is important.

When the irregularity index of a prime p is 1 (as it is for p = 37, 59,
67,...), then there is one progression n = n0 + k*p*(p-1)/2 that gives all
the values of n.

When the irregularity index of a prime p is 2 (as it is for p = 157, 353,
379,...), then there are two progressions n = n0 + d1*k and  n = n0 + d2*k
that give all the values of n.  For p = 157, they are n = 4789 + 24*p*k and
n = 4789 + 54*p*k.

Similarly, when the irregularity index of a prime is r, then there are r
progressions.

Now we just have to prove this conjectured behavior....

Tony