This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A251782 #37 May 14 2024 17:50:46
%S A251782 32,284,37580,1072544,55777784,325656968,42764158652,2444284077476,
%T A251782 46872402575720,4093248733492712,167845040875289732,
%U A251782 4841789050865438960,235423026877046134208,7818983737604766777920,95503904455394036720840,6908622244227620311285724,114945213060615779807957456
%N A251782 Least even integer k such that numerator(B_k) == 0 (mod 37^n).
%C A251782 37 is the first irregular prime. The corresponding entry for the second irregular prime 59 is A299466, and for the third irregular prime 67 is A299467.
%C A251782 The p-adic digits of the unique simple zero of the p-adic zeta function zeta_{(p,l)} with (p,l)=(37,32) were used to compute the sequence (see the Mathematica program below). This corresponds with Table A.2 in Kellner (2007). The sequence is increasing, but some consecutive entries are identical, e.g., entries 18 / 19 and 80 / 81. This is caused only by those p-adic digits that are zero.
%H A251782 Bernd C. Kellner and Robert G. Wilson v, <a href="/A251782/b251782.txt">Table of n, a(n) for n = 1..100</a>
%H A251782 Bernd C. Kellner, <a href="http://bernoulli.org/">The Bernoulli Number Page</a>
%H A251782 Bernd C. Kellner, <a href="https://doi.org/10.1090/S0025-5718-06-01887-4">On irregular prime power divisors of the Bernoulli numbers</a>, Math. Comp. 76 (2007), 405-441; arXiv:<a href="https://arxiv.org/abs/math/0409223">0409223</a> [math.NT], 2004.
%F A251782 Numerator(B_{a(n)}) == 0 (mod 37^n).
%e A251782 a(3) = 37580 because the numerator of B_37580 is divisible by 37^3 and there is no even integer less than 37580 for which this is the case.
%t A251782 p = 37; l = 32; LD = {7, 28, 21, 30, 4, 17, 26, 13, 32, 35, 27, 36, 32, 10, 21, 9, 11, 0, 1, 13, 6, 8, 10, 11, 10, 11, 32, 13, 30, 10, 6, 8, 2, 12, 1, 8, 2, 5, 3, 10, 19, 8, 4, 7, 19, 27, 33, 29, 29, 11, 2, 23, 8, 34, 5, 8, 35, 35, 13, 31, 29, 6, 7, 22, 13, 29, 7, 15, 22, 20, 19, 29, 2, 14, 2, 2, 31, 11, 4, 0, 27, 8, 10, 23, 17, 35, 15, 32, 22, 14, 7, 18, 8, 3, 27, 35, 33, 31, 6}; CalcIndex[L_, p_, l_, n_] := l + (p - 1) Sum[L[[i + 1]] p^i , {i, 0, n - 2}]; Table[CalcIndex[LD, p, l, n], {n, 1, Length[LD] + 1}] // TableForm
%Y A251782 Cf. 2*A091216, 2*A092230, A189683, A299466, A299467.
%K A251782 nonn
%O A251782 1,1
%A A251782 _Bernd C. Kellner_ and _Robert G. Wilson v_, Dec 08 2014
%E A251782 Edited for consistency with A299466 and A299467 by _Bernd C. Kellner_ and _Jonathan Sondow_, Feb 20 2018