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%I #17 Oct 09 2019 20:57:03
%S 58,3292,153640,12597148,846312184,52715297638,320040068824,
%T 370475739904372,23170872799129498,532379740455157312,
%U 111861518490094080436,1314934469494256636776,291496130251698265225984,7852328398132458266800348,1925603427201316655808983674
%N Least even integer k such that numerator(B_k) == 0 (mod 67^n).
%C 67 is the third irregular prime. The corresponding entry for the first irregular prime 37 is A251782, and for the second irregular prime 59 is A299466.
%C The p-adic digits of the unique simple zero of the p-adic zeta function zeta_{(p,l)} with (p,l)=(67,58) 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 22 / 23 and 84 / 85. This is caused only by those p-adic digits that are zero.
%H Bernd C. Kellner, <a href="/A299467/b299467.txt">Table of n, a(n) for n = 1..100</a>
%H Bernd C. Kellner, <a href="http://www.bernoulli.org/">The Bernoulli Number Page</a>
%H Bernd C. Kellner, <a href="http://dx.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.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Regular_prime#Irregular_pairs"> Irregular pairs</a>
%F Numerator(B_{a(n)}) == 0 (mod 67^n).
%e a(3) = 153640 because the numerator of B_153640 is divisible by 67^3 and there is no even integer less than 153640 for which this is the case.
%t p = 67; l = 58; LD = {49, 34, 42, 42, 39, 3, 62, 57, 19, 62, 10, 36, 14, 53, 57, 16, 60, 22, 41, 21, 25, 0, 56, 21, 24, 52, 33, 28, 51, 34, 60, 8, 47, 39, 42, 33, 14, 66, 50, 48, 45, 28, 61, 50, 27, 8, 30, 59, 32, 15, 3, 1, 54, 12, 30, 20, 14, 12, 10, 49, 33, 49, 54, 13, 26, 42, 8, 58, 12, 63, 19, 16, 48, 15, 2, 13, 1, 23, 2, 44, 64, 25, 40, 0, 16, 58, 44, 31, 62, 47, 61, 46, 9, 2, 50, 1, 62, 34, 31}; 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 Cf. 2*A091216, 2*A092230, A189683, A251782, A299466.
%K nonn
%O 1,1
%A _Bernd C. Kellner_ and _Jonathan Sondow_, Feb 15 2018