A299467 Least even integer k such that numerator(B_k) == 0 (mod 67^n).

58, 3292, 153640, 12597148, 846312184, 52715297638, 320040068824, 370475739904372, 23170872799129498, 532379740455157312, 111861518490094080436, 1314934469494256636776, 291496130251698265225984, 7852328398132458266800348, 1925603427201316655808983674

Offset

1,1

Comments

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.

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.

Links

Bernd C. Kellner, 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.

Wikipedia, Irregular pairs

Formula

Numerator(B_{a(n)}) == 0 (mod 67^n).

Example

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.

Mathematica

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

Crossrefs

Cf. 2*A091216, 2*A092230, A189683, A251782, A299466.

Sequence in context: A281031 A225352 A278097 * A334935 A207278 A207474

Adjacent sequences: A299464 A299465 A299466 * A299468 A299469 A299470

Keyword

nonn

Author

Bernd C. Kellner and Jonathan Sondow, Feb 15 2018

Status

approved