A299469 p-adic digits of the unique simple zero of the p-adic zeta-function zeta_{(p,l)} with (p,l) = (59,44).

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

Offset

0,1

Comments

The p-adic digits are used to compute A299466(n) = least even integer k such that numerator(B_k) == 0 (mod 59^n) (see 2nd formula below and the program in A299466).

The algorithm used in the Mathematica program below is from Kellner 2007, Prop. 5.3, p. 428.

The corresponding sequences for (p,l) = (37,32) and (p,l) = (67,58) are A299468 and A299470, respectively.

Links

Bernd C. Kellner and Jonathan Sondow, Table of n, a(n) for n = 0..98

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.

Formula

0 <= a(n) <= 58.

l + (p - 1)*Sum_{i=0..n-2} a(i)*p^i = A299466(n) with (p,l) = (59,44).

Example

The zero is given by a(0) + a(1)*p + a(2)*p^2 + ... with p = 59.

Mathematica

n = 99; p = 59; l = 44;
ModR[x_, m_] := Mod[Mod[Numerator[x], m] PowerMod[Denominator[x], -1, m], m];
B[n_] := -(1 - p^(n - 1)) BernoulliB[n]/n;
T[r_, k_, x_] := Sum[(-1)^(j + k) Binomial[j, k] Binomial[x, j], {j, k, r}];
zt = Table[ModR[B[l + (p - 1) k]/p, p^n], {k, 0, n}];
Z[n_] := zt[[n + 1]]; d = Mod[Z[0] - Z[1], p]; t = 0; L = {};
For[r = 1, r <= n, r++, x = Mod[Sum[Z[k] T[r, k, t], {k, 0, r}], p^r];
  s = ModR[x/(d*p^(r - 1)), p]; AppendTo[L, s]; t += s*p^(r - 1)];
Print[L]

Crossrefs

Cf. A251782, A299466, A299467, A299468, A299470.

Sequence in context: A069823 A133321 A031888 * A046291 A354959 A280406

Adjacent sequences: A299466 A299467 A299468 * A299470 A299471 A299472

Keyword

nonn

Author

Bernd C. Kellner and Jonathan Sondow, Apr 08 2018

Status

approved