A251782 Least even integer k such that numerator(B_k) == 0 (mod 37^n).

32, 284, 37580, 1072544, 55777784, 325656968, 42764158652, 2444284077476, 46872402575720, 4093248733492712, 167845040875289732, 4841789050865438960, 235423026877046134208, 7818983737604766777920, 95503904455394036720840, 6908622244227620311285724, 114945213060615779807957456

Offset: 1

Author

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

Keywords

• nonn

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.

Examples

			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.
		

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, Math. Comp. 76 (2007), 405-441; arXiv:0409223 [math.NT], 2004.

Crossrefs

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

Programs

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

Formula

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

Extensions

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