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A299470
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p-adic digits of the unique simple zero of the p-adic zeta-function zeta_{(p,l)} with (p,l) = (67,58).
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3
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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
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OFFSET
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0,1
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COMMENTS
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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.
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LINKS
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FORMULA
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0 <= a(n) <= 66.
l + (p - 1)*Sum_{i=0..n-2} a(i)*p^i = A299467(n) with (p,l) = (67,58).
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EXAMPLE
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The zero is given by a(0) + a(1)*p + a(2)*p^2 + ... with p = 67.
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MATHEMATICA
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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]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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