[seqfan] Re: Computing more terms of draft sequence A343745

[Neil Sloane]

Tom, Thanks for giving a clearer definition.

The sequence of irregular primes is A000928, and it has programs in the
usual languages. The b-file has 10000 terms, the 10000-th being 294001. No
reason we can't have a 20K-term b-file, and/or a 100K-term a-file.

Felix just submitted the two terms 491 and 587 (as A343745), but I said we
needed more terms.

Best regards
Neil

Neil J. A. Sloane, President, OEIS Foundation.
11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
Phone: 732 828 6098; home page: http://NeilSloane.com
Email: njasloane at gmail.com



On Fri, Apr 30, 2021 at 9:04 PM Tom Duff <eigenvectors at gmail.com> wrote:

> The Wells Johnson paper is in Math Comp (Jan 1975, pp 113-120). JSTOR
> has it. In the introduction he says
>
>     A prime p is called regular if it does not divide the numerator of
> any of the Bernoulli numbers B_2, B_4, ... , B_{p-1}.
>
> and later
>
>    If p is an irregular prime and p divides the numerator of the
> Bernoulli number B_{2k} for 0<2k<p-1, we shall refer to (p,2k) as an
> irregular pair.
>
> Still later we have
>
>    It has been known for a long time that consecutive irregular pairs
> (those of the form (p,2k) and (p, 2k+2)) occur for p=491 and 587.
>
> So it looks like you only have to check that prime p divides B_{2k}
> for some 2k<p-1 to determine that p is an irregular prime, and then if
> p divides both B_{2k} and B_{2k+2} (again for some 2k<p-1, or more
> likely 2k<p-3 -- it's not 100% clear) we have consecutive irregular
> prime pairs.
>
> It looks like the name of the sequence should be
>
> Consecutive irregular *prime* pairs, i.e., primes p where an integer k
> exists such that p divides the numerators of the Bernoulli numbers
> B_{2k} and B_{2k+2}, *where 2k+2<p-1*.
> (changes marked with *...*)
>
> On Fri, Apr 30, 2021 at 4:42 PM D. S. McNeil <dsm054 at gmail.com> wrote:
> >
> > I think the 'order in which they appear' definition is going to be more
> > practical than the pure "this is the set of primes that satisfy the
> > condition" case, because otherwise it might take a good bit of effort to
> > prove that different primes aren't in the set.
> >
> > The numbers generated by a quick scan I did match rgwv's, so I think
> we're
> > interpreting the definition in the same way.  OTOH something must be
> wrong
> > somewhere, because 37 shows up at such a small k that it should've been
> > caught.
> >
> > Is there an additional condition on p which rules out numbers like 37 and
> > 103?
> >
> >
> > Doug
> >
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