A299467 -id:A299467 - cp's OEIS frontend

A251782 Least even integer k such that numerator(B_k) == 0 (mod 37^n).

32, 284, 37580, 1072544, 55777784, 325656968, 42764158652, 2444284077476, 46872402575720, 4093248733492712, 167845040875289732, 4841789050865438960, 235423026877046134208, 7818983737604766777920, 95503904455394036720840, 6908622244227620311285724, 114945213060615779807957456

Offset: 1

Bernd C. Kellner and Robert G. Wilson v, Dec 08 2014

nonn

37 is the first irregular prime. The corresponding entry for the second irregular prime 59 is A299466, and for the third irregular prime 67 is A299467.

The p-adic digits of the unique simple zero of the p-adic zeta function zeta_{(p,l)} with (p,l)=(37,32) were used to compute the sequence (see the Mathematica program below). This corresponds with Table A.2 in Kellner (2007). The sequence is increasing, but some consecutive entries are identical, e.g., entries 18 / 19 and 80 / 81. This is caused only by those p-adic digits that are zero.

			a(3) = 37580 because the numerator of B_37580 is divisible by 37^3 and there is no even integer less than 37580 for which this is the case.

Bernd C. Kellner and Robert G. Wilson v, Table of n, a(n) for n = 1..100
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; arXiv:0409223 [math.NT], 2004.

Cf. 2*A091216, 2*A092230, A189683, A299466, A299467.

p = 37; l = 32; LD = {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}; CalcIndex[L_, p_, l_, n_] := l + (p - 1) Sum[L[[i + 1]] p^i , {i, 0, n - 2}]; Table[CalcIndex[LD, p, l, n], {n, 1, Length[LD] + 1}] // TableForm

Numerator(B_{a(n)}) == 0 (mod 37^n).

Edited for consistency with A299466 and A299467 by Bernd C. Kellner and Jonathan Sondow, Feb 20 2018

A299466 Least even integer k such that numerator(B_k) == 0 (mod 59^n).

Original entry on oeis.org

44, 914, 86464, 8162384, 436993736, 13087518620, 469209221382, 42059215391408, 4083629226737464, 498021221327673308, 5020105038665551466, 1516903461301962815624, 24254443348634296180510, 2604090699795956735657960, 252229046873638875979496022

Offset: 1

Bernd C. Kellner and Jonathan Sondow, Feb 10 2018

nonn

59 is the second irregular prime. The corresponding entry for the first irregular prime 37 is A251782, and for the third irregular prime 67 is A299467.

The p-adic digits of the unique simple zero of the p-adic zeta function zeta_{(p,l)} with (p,l)=(59,44) were used to compute the sequence (see the Mathematica program below). This corresponds with Table A.2 in Kellner (2007). The sequence is increasing, but some consecutive entries are identical, e.g., entries 30 / 31 and 94 / 95. This is caused only by those p-adic digits that are zero.

			a(3) = 86464 because the numerator of B_86464 is divisible by 59^3 and there is no even integer less than 86464 for which this is the case.

Bernd C. Kellner, Table of n, a(n) for n = 1..100
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.
Wikipedia, Irregular pairs

Cf. 2*A091216, 2*A092230, A189683, A251782, A299467.

p = 59; l = 44; LD = {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}; CalcIndex[L_, p_, l_, n_] := l + (p - 1) Sum[L[[i + 1]] p^i , {i, 0, n -2}]; Table[CalcIndex[LD, p, l, n], {n, 1, Length[LD] + 1}] // TableForm

Numerator(B_{a(n)}) == 0 (mod 59^n).

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

Original entry on oeis.org

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

Bernd C. Kellner and Jonathan Sondow, Mar 28 2018

nonn

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.

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

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.

Cf. A251782, A299466, A299467, A299469, A299470.

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]

0 <= a(n) <= 36.

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

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

Original entry on oeis.org

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

Bernd C. Kellner and Jonathan Sondow, Apr 08 2018

nonn

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.

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

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.

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

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]

0 <= a(n) <= 58.

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

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

Original entry on oeis.org

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

Bernd C. Kellner and Jonathan Sondow, Apr 08 2018

nonn

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.

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

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.

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

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]

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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A251782 Least even integer k such that numerator(B_k) == 0 (mod 37^n).

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A299466 Least even integer k such that numerator(B_k) == 0 (mod 59^n).

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

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

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A299470 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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Examples

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Crossrefs

Programs

Mathematica

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