A091216 -id:A091216 - cp's OEIS frontend

A092221 Numbers k such that numerator of Bernoulli(2*k) is divisible by 59, the second irregular prime.

22, 51, 59, 80, 109, 118, 138, 167, 177, 196, 225, 236, 254, 283, 295, 312, 341, 354, 370, 399, 413, 428, 457, 472, 486, 515, 531, 544, 573, 590, 602, 631, 649, 660, 689, 708, 718, 747, 767, 776, 805, 826, 834, 863, 885, 892, 921, 944, 950, 979, 1003, 1008

Offset: 1

Robert G. Wilson v, Feb 25 2004

nonn

Satisfies a(n) = 2*a(n-3) - a(n-6) for n < 67. - Chai Wah Wu, May 28 2016

Amiram Eldar, Table of n, a(n) for n = 1..6450 (terms 1..1000 from Joerg Arndt)
Eric Weisstein's World of Mathematics, Bernoulli Number.

Cf. A000928, A091216, A092222, A092223, A092224, A092225, A092226, A092227, A092228, A092229, A092231.

Select[ Range[ 1036], Mod[ Numerator[ BernoulliB[2# ]], 59] == 0 &]

for(n=0, 10^3, if( numerator(bernfrac(2*n))%59==0, print1(n, ", ") ) ); \\ Joerg Arndt, May 29 2016

from sympy import bernoulli
A092221_list = [n for n in range(10**3) if not bernoulli(2*n).p % 59] # Chai Wah Wu, May 28 2016

A092229 Numbers k such that numerator of Bernoulli(2*k) is divisible by 257, the tenth irregular prime.

Original entry on oeis.org

82, 210, 257, 338, 466, 514, 594, 722, 771, 850, 978, 1028, 1106, 1234, 1285, 1362, 1490, 1542, 1618, 1746, 1799, 1874, 2002, 2056, 2130, 2258, 2313, 2386, 2514, 2570, 2642, 2770, 2827, 2898, 3026, 3084, 3154, 3282, 3341, 3410, 3538, 3598, 3666, 3794

Offset: 1

Robert G. Wilson v, Feb 25 2004

nonn

Amiram Eldar, Table of n, a(n) for n = 1..1500
Eric Weisstein's World of Mathematics, Bernoulli Number.

Cf. A000928, A091216, A092221, A092222, A092223, A092224, A092225, A092226, A092227, A092228.

Select[ Range[ 3854], Mod[ Numerator[ BernoulliB[2# ]], 257] == 0 &]

A092222 Numbers k such that numerator of Bernoulli(2*k) is divisible by 67, the third irregular prime.

Original entry on oeis.org

29, 62, 67, 95, 128, 134, 161, 194, 201, 227, 260, 268, 293, 326, 335, 359, 392, 402, 425, 458, 469, 491, 524, 536, 557, 590, 603, 623, 656, 670, 689, 722, 737, 755, 788, 804, 821, 854, 871, 887, 920, 938, 953, 986, 1005, 1019, 1052, 1072, 1085, 1118, 1139

Offset: 1

Robert G. Wilson v, Feb 25 2004

nonn

n is a member iff either 2n == 58 (mod 66) or n = 67k. - T. D. Noe, Mar 22 2004

Amiram Eldar, Table of n, a(n) for n = 1..5700
Eric Weisstein's World of Mathematics, Bernoulli Number.

Cf. A000928, A091216, A092221, A092223, A092224, A092225, A092226, A092227, A092228, A092229.

Select[ Range[ 1150], Mod[ Numerator[ BernoulliB[2# ]], 67] == 0 &]

A092223 Numbers k such that numerator of Bernoulli(2*k) is divisible by 101, the fourth irregular prime.

Original entry on oeis.org

34, 84, 101, 134, 184, 202, 234, 284, 303, 334, 384, 404, 434, 484, 505, 534, 584, 606, 634, 684, 707, 734, 784, 808, 834, 884, 909, 934, 984, 1010, 1034, 1084, 1111, 1134, 1184, 1212, 1234, 1284, 1313, 1334, 1384, 1414, 1434, 1484, 1515, 1534, 1584, 1616

Offset: 1

Robert G. Wilson v, Feb 25 2004

nonn

Amiram Eldar, Table of n, a(n) for n = 1..3800
Eric Weisstein's World of Mathematics, Bernoulli Number.

Cf. A000928, A091216, A092221, A092222, A092224, A092225, A092226, A092227, A092228, A092229.

Select[ Range[ 1633], Mod[ Numerator[ BernoulliB[2# ]], 101] == 0 &]

A092224 Numbers k such that the numerator of Bernoulli(2*k) is divisible by 103, the fifth irregular prime.

Original entry on oeis.org

12, 63, 103, 114, 165, 206, 216, 267, 309, 318, 369, 412, 420, 471, 515, 522, 573, 618, 624, 675, 721, 726, 777, 824, 828, 879, 927, 930, 981, 1030, 1032, 1083, 1133, 1134, 1185, 1236, 1287, 1338, 1339, 1389, 1440, 1442, 1491, 1542, 1545, 1593, 1644, 1648

Offset: 1

Robert G. Wilson v, Feb 25 2004

nonn

103 = A094095(1) is the first irregular prime in A094095. This sequence is the union of 2 arithmetic progressions: (24 + 102*n)/2 and 103*n. Note that the numerator of BernoulliB(2*114) is divisible by the first nontrivial irregular squared prime 103^2, when A090943(1)/2 = a(n) = 114 = (24 + 102*2)/2. Also, the numerator of BernoulliB(2*1236) is divisible by 103^2 because a(n) = 1236 = (24 + 102*24)/2 = 103*24/2. - Alexander Adamchuk, Jul 31 2006

Amiram Eldar, Table of n, a(n) for n = 1..3700
Eric Weisstein's World of Mathematics, Bernoulli Number.

Cf. A000928, A091216, A092221, A092222, A092223, A092225, A092226, A092227, A092228, A092229.

Cf. A094095, A090943, A027641.

Select[ Range[ 1694], Mod[ Numerator[ BernoulliB[2# ]], 103] == 0 &]
Select[Union[Table[2n*103,{n,1,100}],Table[24+102*n,{n,0,100}]], #<=10000&]/2 (* Alexander Adamchuk, Jul 31 2006 *)

A092225 Numbers k such that numerator of Bernoulli(2*k) is divisible by 131, the sixth irregular prime.

Original entry on oeis.org

11, 76, 131, 141, 206, 262, 271, 336, 393, 401, 466, 524, 531, 596, 655, 661, 726, 786, 791, 856, 917, 921, 986, 1048, 1051, 1116, 1179, 1181, 1246, 1310, 1311, 1376, 1441, 1506, 1571, 1572, 1636, 1701, 1703, 1766, 1831, 1834, 1896, 1961, 1965, 2026, 2091

Offset: 1

Robert G. Wilson v, Feb 25 2004

nonn

Amiram Eldar, Table of n, a(n) for n = 1..2900
Eric Weisstein's World of Mathematics, Bernoulli Number.

Cf. A000928, A091216, A092221, A092222, A092223, A092224, A092226, A092227, A092228, A092229.

Select[ Range[ 2095], Mod[ Numerator[ BernoulliB[2# ]], 131] == 0 &]

A092226 Numbers k such that numerator of Bernoulli(2*k) is divisible by 149, the seventh irregular prime.

Original entry on oeis.org

65, 139, 149, 213, 287, 298, 361, 435, 447, 509, 583, 596, 657, 731, 745, 805, 879, 894, 953, 1027, 1043, 1101, 1175, 1192, 1249, 1323, 1341, 1397, 1471, 1490, 1545, 1619, 1639, 1693, 1767, 1788, 1841, 1915, 1937, 1989, 2063, 2086, 2137, 2211, 2235, 2285

Offset: 1

Robert G. Wilson v, Feb 25 2004

nonn

Amiram Eldar, Table of n, a(n) for n = 1..2500
Eric Weisstein's World of Mathematics, Bernoulli Number.

Cf. A000928, A091216, A092221, A092222, A092223, A092224, A092225, A092227, A092228, A092229.

Select[ Range[ 2358], Mod[ Numerator[ BernoulliB[2# ]], 149] == 0 &]

A092227 Numbers k such that numerator of Bernoulli(2*k) is divisible by 157, the eighth irregular prime and first irregular prime of index 2.

Original entry on oeis.org

31, 55, 109, 133, 157, 187, 211, 265, 289, 314, 343, 367, 421, 445, 471, 499, 523, 577, 601, 628, 655, 679, 733, 757, 785, 811, 835, 889, 913, 942, 967, 991, 1045, 1069, 1099, 1123, 1147, 1201, 1225, 1256, 1279, 1303, 1357, 1381, 1413, 1435, 1459, 1513, 1537

Offset: 1

Robert G. Wilson v, Feb 25 2004

nonn

Amiram Eldar, Table of n, a(n) for n = 1..4000
Eric Weisstein's World of Mathematics, Bernoulli Number.

Cf. A000928, A073277, A091216, A092221, A092222, A092223, A092224, A092225, A092226, A092228, A092229.

Select[ Range[ 1569], Mod[ Numerator[ BernoulliB[2# ]], 157] == 0 &]

A092228 Numbers k such that numerator of Bernoulli(2*k) is divisible by 233, the ninth irregular prime.

Original entry on oeis.org

42, 158, 233, 274, 390, 466, 506, 622, 699, 738, 854, 932, 970, 1086, 1165, 1202, 1318, 1398, 1434, 1550, 1631, 1666, 1782, 1864, 1898, 2014, 2097, 2130, 2246, 2330, 2362, 2478, 2563, 2594, 2710, 2796, 2826, 2942, 3029, 3058, 3174, 3262, 3290, 3406, 3495

Offset: 1

Robert G. Wilson v, Feb 25 2004

nonn

Amiram Eldar, Table of n, a(n) for n = 1..1600
Eric Weisstein's World of Mathematics, Bernoulli Number.

Cf. A000928, A091216, A092221, A092222, A092223, A092224, A092225, A092226, A092227, A092229.

Select[ Range[ 3521], Mod[ Numerator[ BernoulliB[2# ]], 233] == 0 &]

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

Original entry on oeis.org

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

cp's OEIS Frontend

A092221 Numbers k such that numerator of Bernoulli(2*k) is divisible by 59, the second irregular prime.

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A092229 Numbers k such that numerator of Bernoulli(2*k) is divisible by 257, the tenth irregular prime.

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A092222 Numbers k such that numerator of Bernoulli(2*k) is divisible by 67, the third irregular prime.

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A092223 Numbers k such that numerator of Bernoulli(2*k) is divisible by 101, the fourth irregular prime.

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A092224 Numbers k such that the numerator of Bernoulli(2*k) is divisible by 103, the fifth irregular prime.

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A092225 Numbers k such that numerator of Bernoulli(2*k) is divisible by 131, the sixth irregular prime.

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A092226 Numbers k such that numerator of Bernoulli(2*k) is divisible by 149, the seventh irregular prime.

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A092227 Numbers k such that numerator of Bernoulli(2*k) is divisible by 157, the eighth irregular prime and first irregular prime of index 2.

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A092228 Numbers k such that numerator of Bernoulli(2*k) is divisible by 233, the ninth irregular prime.

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

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