OFFSET
0,1
COMMENTS
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) <= 36.
l + (p - 1)*Sum_{i=0..n-2} a(i)*p^i = A251782(n) with (p,l) = (37,32).
EXAMPLE
The zero is given by a(0) + a(1)*p + a(2)*p^2 + ... with p = 37.
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]
CROSSREFS
KEYWORD
nonn
AUTHOR
Bernd C. Kellner and Jonathan Sondow, Mar 28 2018
STATUS
approved