A299466 Least even integer k such that numerator(B_k) == 0 (mod 59^n).

44, 914, 86464, 8162384, 436993736, 13087518620, 469209221382, 42059215391408, 4083629226737464, 498021221327673308, 5020105038665551466, 1516903461301962815624, 24254443348634296180510, 2604090699795956735657960, 252229046873638875979496022

Offset

1,1

Comments

59 is the second irregular prime. The corresponding entry for the first irregular prime 37 is A251782, 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)=(59,44) 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 30 / 31 and 94 / 95. 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 59^n).

Example

a(3) = 86464 because the numerator of B_86464 is divisible by 59^3 and there is no even integer less than 86464 for which this is the case.

Mathematica

p = 59; l = 44; LD = {15, 25, 40, 36, 18, 11, 17, 28, 58, 9, 51, 13, 25, 41, 44, 17, 43, 35, 21, 10, 21, 38, 9, 12, 40, 43, 45, 30, 41, 0, 3, 25, 34, 49, 45, 9, 19, 48, 57, 11, 13, 29, 28, 44, 41, 37, 33, 29, 43, 8, 57, 12, 48, 15, 15, 53, 57, 16, 51, 16, 54, 30, 9, 26, 8, 49, 22, 58, 11, 42, 28, 36, 33, 45, 24, 32, 18, 12, 29, 45, 40, 27, 19, 40, 41, 11, 42, 49, 35, 41, 57, 54, 33, 0, 34, 34, 49, 6, 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, A299467.

Sequence in context: A133349 A010838 A191374 * A010960 A035717 A035611

Adjacent sequences: A299463 A299464 A299465 * A299467 A299468 A299469

Keyword

nonn

Author

Bernd C. Kellner and Jonathan Sondow, Feb 10 2018

Status

approved