A299470 p-adic digits of the unique simple zero of the p-adic zeta-function zeta_{(p,l)} with (p,l) = (67,58).

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

Offset: 0

Author

Bernd C. Kellner and Jonathan Sondow, Apr 08 2018

Keywords

• nonn

Comments

The p-adic digits are used to compute A299467(n) = least even integer k such that numerator(B_k) == 0 (mod 67^n) (see 2nd formula below and the program in A299467).

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) = (59,44) are A299468 and A299469, respectively.

Examples

			The zero is given by a(0) + a(1)*p + a(2)*p^2 + ... with p = 67.
		

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.

Crossrefs

Cf. A251782, A299466, A299467, A299468, A299469.

Programs

Mathematica

n = 99; p = 67; l = 58;
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]

Formula

0 <= a(n) <= 66.

l + (p - 1)*Sum_{i=0..n-2} a(i)*p^i = A299467(n) with (p,l) = (67,58).