A092229 -id:A092229 - cp's OEIS frontend

A091216 Numbers k such that numerator of Bernoulli(2*k) is divisible by 37, the first irregular prime.

16, 34, 37, 52, 70, 74, 88, 106, 111, 124, 142, 148, 160, 178, 185, 196, 214, 222, 232, 250, 259, 268, 286, 296, 304, 322, 333, 340, 358, 370, 376, 394, 407, 412, 430, 444, 448, 466, 481, 484, 502, 518, 520, 538, 555, 556, 574, 592, 610, 628

Offset: 1

N. J. A. Sloane, Feb 24 2004

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..100 from Seiichi Manyama)

Cf. A000928, A092221, A092222, A092223, A092224, A092225, A092226, A092227, A092228, A092229, A092230, A092231.

Select[ Range[ 681], Mod[ Numerator[ BernoulliB[2# ]], 37] == 0 &] (* Robert G. Wilson v, Feb 24 2004 *)

for(j=1,260, if (! (numerator(bernfrac(2*j))%37), print1(j, ", ")))

More terms from Robert G. Wilson v, Feb 24 2004

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

Original entry on oeis.org

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

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 &]

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

A091216 Numbers k such that numerator of Bernoulli(2*k) is divisible by 37, the first irregular prime.

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