A091216 -id:A091216 - cp's OEIS frontend

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

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).

A299467 Least even integer k such that numerator(B_k) == 0 (mod 67^n).

Original entry on oeis.org

58, 3292, 153640, 12597148, 846312184, 52715297638, 320040068824, 370475739904372, 23170872799129498, 532379740455157312, 111861518490094080436, 1314934469494256636776, 291496130251698265225984, 7852328398132458266800348, 1925603427201316655808983674

Offset: 1

Bernd C. Kellner and Jonathan Sondow, Feb 15 2018

nonn

67 is the third irregular prime. The corresponding entry for the first irregular prime 37 is A251782, and for the second irregular prime 59 is A299466.

The p-adic digits of the unique simple zero of the p-adic zeta function zeta_{(p,l)} with (p,l)=(67,58) 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 22 / 23 and 84 / 85. This is caused only by those p-adic digits that are zero.

			a(3) = 153640 because the numerator of B_153640 is divisible by 67^3 and there is no even integer less than 153640 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, A299466.

p = 67; l = 58; LD = {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}; 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 67^n).

A092230 Numbers k such that the numerator of Bernoulli(2k) is divisible by the square of 37, the first irregular prime.

Original entry on oeis.org

142, 592, 808, 1258, 1369, 1474, 1924, 2140, 2590, 2738, 2806, 3256, 3472, 3922, 4107, 4138, 4588, 4804, 5254, 5470, 5476, 5920, 6136, 6586, 6802, 6845, 7252, 7468, 7918, 8134, 8214, 8584, 8800, 9250, 9466, 9583, 9916, 10132, 10582, 10798, 10952, 11248

Offset: 1

Robert G. Wilson v, Feb 25 2004

nonn

Eric Weisstein's World of Mathematics, Bernoulli Number.

Equals A090789/2. Cf. A000928, A091216.

Select[ Range[ 5780], Mod[ Numerator[ BernoulliB[2# ]], 37^2] == 0 &]

See A090789.

More terms from T. D. Noe, Feb 26 2004

A092231 Numbers k such that numerator of Bernoulli(2*k) is divisible by 37 and 59, the first two irregular primes.

Original entry on oeis.org

196, 370, 718, 826, 1240, 1443, 1762, 1888, 2183, 2284, 2516, 2806, 2950, 3328, 3589, 3850, 4012, 4366, 4372, 4894, 5074, 5416, 5735, 5938, 6136, 6460, 6549, 6808, 6982, 7198, 7504, 7881, 8026, 8260, 8548, 8732, 8954, 9070, 9322, 9592, 10027, 10114

Offset: 1

Robert G. Wilson v, Feb 25 2004

nonn

Amiram Eldar, Table of n, a(n) for n = 1..500
Eric Weisstein's World of Mathematics, Bernoulli Number.

Intersection of A091216 and A092221.

Cf. A000928.

Select[ Range[ 5780], Mod[ Numerator[ BernoulliB[ 2# ]], 37] == Mod[ Numerator[ BernoulliB[ 2# ]], 59] == 0 &]

A093058 Numbers k such that numerator of Bernoulli(2k) is divisible by the square of 59, the second irregular prime.

Original entry on oeis.org

457, 1298, 2168, 3009, 3481, 3879, 4720, 5590, 6431, 6962, 7301, 8142, 9012, 9853, 10443, 10723, 11564, 12434, 13275, 13924, 14145, 14986, 15856, 16697, 17405, 17567, 18408

Offset: 1

Robert G. Wilson v, Feb 26 2004

nonn

Eric Weisstein's World of Mathematics, Bernoulli Number.

Cf. A000928, A091216, A092221.

Select[ Range[ 9695], Mod[ Numerator[ BernoulliB[2# ]], 59^2] == 0 &]

If we omit multiples of 3481 and take first differences, it appears that we get a common difference of {841, 870} repeated.

More terms from Eric W. Weisstein, Mar 19 2004

A093059 Numbers k such that the numerator of Bernoulli(2k) is divisible by the square of 67, the third irregular prime.

Original entry on oeis.org

1646, 1943, 3857, 4154, 4489, 6068, 6365, 8279, 8576, 8978, 10490, 10787, 12701, 12998, 13467, 14912, 15209, 17123, 17420, 17956

Offset: 1

Robert G. Wilson v, Feb 26 2004

nonn

Eric Weisstein's World of Mathematics, Bernoulli Number.

Cf. A000928, A091216, A092221.

Select[ Range[ 9695], Mod[ Numerator[ BernoulliB[2# ]], 67^2] == 0 &]

More terms from Eric W. Weisstein, Mar 19 2004

A281502 Numbers m such that the numerator of Bernoulli(2m) is divisible by 691.

Original entry on oeis.org

6, 100, 351, 445, 691, 696, 790, 1041, 1135, 1382, 1386, 1480, 1731, 1825, 2073, 2076, 2170, 2421, 2515, 2764, 2766, 2860, 3111, 3205, 3455, 3456, 3550, 3801, 3895, 4146, 4240, 4491, 4585, 4836, 4837, 4930, 5181, 5275, 5526, 5528, 5620, 5871, 5965

Offset: 1

Seiichi Manyama, Jan 23 2017

nonn

6 + 345*k and 100 + 345*k are terms for k >= 0.

			Bernoulli(2*6) = -691/2730. So 6 is a term.

Bernd C. Kellner, The Bernoulli Number Page.
Eric Weisstein's World of Mathematics, Bernoulli Number
Wikipedia, Kummer's congruences

Cf. A000928, A091216, A092221 - A092229, A119864.

Select[Range[4930],Mod[Numerator[BernoulliB[2#]],  691] == 0 &] (* Indranil Ghosh, Mar 11 2017 *)

is(n) = Mod(numerator(bernfrac(2*n)), 691)==0 \\ Felix Fröhlich, Jan 23 2017

from itertools import count, islice
from sympy import bernoulli
def A281502gen(): return filter(lambda n:not bernoulli(2*n).p % 691,count(0))
A281502_list = list(islice(A281502gen(),20)) # Chai Wah Wu, Dec 21 2021

a(n) = A119864(n)/2.

a(12) - a(36) from Seiichi Manyama, Jan 24 2017

More terms from Indranil Ghosh, Mar 11 2017

cp's OEIS Frontend

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

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A299467 Least even integer k such that numerator(B_k) == 0 (mod 67^n).

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A092230 Numbers k such that the numerator of Bernoulli(2k) is divisible by the square of 37, the first irregular prime.

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A092231 Numbers k such that numerator of Bernoulli(2*k) is divisible by 37 and 59, the first two irregular primes.

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A093058 Numbers k such that numerator of Bernoulli(2k) is divisible by the square of 59, the second irregular prime.

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A093059 Numbers k such that the numerator of Bernoulli(2k) is divisible by the square of 67, the third irregular prime.

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A281502 Numbers m such that the numerator of Bernoulli(2m) is divisible by 691.

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