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A251782 Least even integer k such that numerator(B_k) == 0 (mod 37^n). 6
32, 284, 37580, 1072544, 55777784, 325656968, 42764158652, 2444284077476, 46872402575720, 4093248733492712, 167845040875289732, 4841789050865438960, 235423026877046134208, 7818983737604766777920, 95503904455394036720840, 6908622244227620311285724, 114945213060615779807957456 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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.

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.

LINKS

Bernd C. Kellner and Robert G. Wilson v, Table of n, a(n) for n = 1..100

Bernd C. Kellner, The Bernoulli Number Page

Bernd C. Kellner, On irregular prime power divisors of the Bernoulli numbers, arXiv:math/0409223 [math.NT], 2004-2005.

Bernd C. Kellner, On irregular prime power divisors of the Bernoulli numbers, Math. Comp. 76 (2007) 405-441.

FORMULA

Numerator(B_{a(n)}) == 0 (mod 37^n).

EXAMPLE

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.

MATHEMATICA

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

CROSSREFS

Cf. 2*A091216, 2*A092230, A189683, A299466, A299467.

Sequence in context: A317151 A316881 A317609 * A267324 A197523 A337418

Adjacent sequences:  A251779 A251780 A251781 * A251783 A251784 A251785

KEYWORD

nonn

AUTHOR

Bernd C. Kellner and Robert G. Wilson v, Dec 08 2014

EXTENSIONS

Edited for consistency with A299466 and A299467 by Bernd C. Kellner and Jonathan Sondow, Feb 20 2018

STATUS

approved

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Last modified October 17 00:35 EDT 2021. Contains 348048 sequences. (Running on oeis4.)