A299468 p-adic digits of the unique simple zero of the p-adic zeta-function zeta_{(p,l)} with (p,l) = (37,32).

7, 28, 21, 30, 4, 17, 26, 13, 32, 35, 27, 36, 32, 10, 21, 9, 11, 0, 1, 13, 6, 8, 10, 11, 10, 11, 32, 13, 30, 10, 6, 8, 2, 12, 1, 8, 2, 5, 3, 10, 19, 8, 4, 7, 19, 27, 33, 29, 29, 11, 2, 23, 8, 34, 5, 8, 35, 35, 13, 31, 29, 6, 7, 22, 13, 29, 7, 15, 22, 20, 19, 29, 2, 14, 2, 2, 31, 11, 4, 0, 27, 8, 10, 23, 17, 35, 15, 32, 22, 14, 7, 18, 8, 3, 27, 35, 33, 31, 6

Offset: 0

Author

Bernd C. Kellner and Jonathan Sondow, Mar 28 2018

Keywords

• nonn

Comments

The p-adic digits are used to compute A251782(n) = least even integer k such that numerator(B_k) == 0 (mod 37^n) (see 2nd formula below and the program in A251782).

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

The corresponding sequences for (p,l) = (59,44) and (p,l) = (67,58) are A299469 and A299470, respectively.

Examples

			The zero is given by a(0) + a(1)*p + a(2)*p^2 + ... with p = 37.
		

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, A299469, A299470.

Programs

Mathematica

n = 99; p = 37; l = 32;
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) <= 36.

l + (p - 1)*Sum_{i=0..n-2} a(i)*p^i = A251782(n) with (p,l) = (37,32).