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 A299466 Least even integer k such that numerator(B_k) == 0 (mod 59^n). 6
 44, 914, 86464, 8162384, 436993736, 13087518620, 469209221382, 42059215391408, 4083629226737464, 498021221327673308, 5020105038665551466, 1516903461301962815624, 24254443348634296180510, 2604090699795956735657960, 252229046873638875979496022 (list; graph; refs; listen; history; text; internal format)
 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

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Last modified February 1 17:02 EST 2023. Contains 359993 sequences. (Running on oeis4.)