A094095 -id:A094095 - cp's OEIS frontend

A090943 Even numbers n such that N(n) is divisible by a nontrivial square, say m^2 with gcd(n,m) = 1, where N(n) is the numerator of the Bernoulli number B(n). The smallest numbers m are given in A094095.

228, 284, 914, 1434, 1616, 2948, 3292, 4280, 4336, 5612, 5768, 6302, 6944, 7714, 7758, 8276, 9608

Offset: 1

T. D. Noe, Feb 27 2004

This sequence consists of the union of an infinite number of arithmetic progressions. Let p be an irregular prime and let {m1, m2, ...} be even numbers < p*(p-1) such that p^2 | N(mi). Then each pair (p, mi) is a second-order irregular pair. This leads to the arithmetic progression n = mi + p*(p-1)*k for each i and for k = 0, 1, 2, 3, ... If we restrict the sequence to those pairs with mi < 10000, we find that only the pairs (37, 284), (59, 914), (67, 3292), (101, 5768), (103, 228), (157, 6302) and (271, 1434) contribute terms to this sequence.

Bernd Kellner, Über irregulaere Paare hoeherer Ordnungen [On irregular pairs of higher order], Diplomarbeit, Goettingen 2002.
S. S. Wagstaff, Jr., Prime divisors of the Bernoulli and Euler numbers, 2018.
Charles Weibel, Algebraic K-Theory of Rings of Integers in Local and Global Fields, in: E. Friedlander and D. Grayson (eds), Handbook of K-Theory, Springer, Berlin, Heidelberg, Vol 1, 2005, pp. 139-190; see Example 96 on p. 180.

Cf. A092681, A094095.

nn=10; s = Union[284 + 36*37*Range[0, nn], 914+58*59*Range[0, nn], 3292+66*67*Range[0, nn], 5768+100*101*Range[0, nn], 228+102*103*Range[0, nn], 6302+156*157*Range[0, nn], 1434+270*271*Range[0, nn]]; Select[s, #<=10000&]

Addition of the word "smallest" in the name by Petros Hadjicostas, May 12 2020

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 *)

A090997 Numbers m such that the numerator of the Bernoulli number B(m) is divisible by a square.

Original entry on oeis.org

50, 98, 150, 196, 228, 242, 250, 284, 338, 350, 392, 450, 484, 490, 550, 578, 650, 676, 686, 722, 726, 750, 784, 850, 914, 950, 968, 980, 1014, 1050, 1058, 1078, 1150, 1156, 1184, 1250, 1274, 1350, 1352, 1372, 1434, 1444, 1450, 1452, 1550, 1568, 1616

Offset: 1

Hans Havermann, Feb 28 2004

nonn

It appears that all terms that are divisible by p^2 and do not belong to A090943 are of the form 2*k*p^2, where p is a prime and k > 0 is an integer. Also, all numbers in A090943 are terms because they are divisible by the squares of irregular primes in A094095. The corresponding smallest primes p such that their squares divide terms are listed in A090987. - Alexander Adamchuk, Aug 19 2006

A subsequence of the current sequence is A122270, which are the numbers m such that the numerator of the Bernoulli number B(m) is divisible by a cube. Another subsequence of the current sequence is A122272, which are the numbers m such that the numerator of the Bernoulli number B(m) is divisible by p^4, where p is a prime. Note that the numerator of the Bernoulli number B(6250) is divisible by 5^5. - Alexander Adamchuk, Aug 28 2006

			a(3) = 150 because numerator(B(150)) == 0 (mod 5^2).

Alexander Adamchuk, Aug 28 2006, Table of n, a(n) for n = 1..152 (term 3886 added by Daniel Suteu)
The Bernoulli Number Page, Table of factors of the numerators of Bernoulli numbers Bn in the range n = 2..10000, 2018.
S. S. Wagstaff, Jr, Prime factors of the absolute values of Bernoulli numerators, 2018.

Cf. A000367, A090943, A094095. For the smallest square factor, see A090987.

Cf. A122270, A122271, A122272, A122273.

In view of the phrase "it appears", it is not clear to me that the correctness of this sequence has been rigorously established. - N. J. A. Sloane, Aug 26 2006
More terms from Alexander Adamchuk, Aug 19 2006
More terms from Alexander Adamchuk, Aug 28 2006
Various sections edited by Petros Hadjicostas, May 12 2020
Incorrect term 294 removed by Daniel Suteu, May 21 2020

A090987 a(n) is the smallest prime whose square divides the numerator of the Bernoulli number B(A090997(n)).

Original entry on oeis.org

5, 7, 5, 7, 103, 11, 5, 37, 13, 5, 7, 5, 11, 7, 5, 17, 5, 13, 7, 19, 11, 5, 7, 5, 59, 5, 11, 7, 13, 5, 23, 7, 5, 17, 37, 5, 7, 5, 13, 7, 271, 19, 5, 11, 5, 7, 37, 5, 7, 29, 13, 11, 17, 5, 5, 7, 31, 11, 5, 7, 5, 23, 5, 7, 19, 11, 5, 7, 17, 5, 13, 5, 103, 37, 7, 5, 59, 5, 11, 13, 37, 7, 5, 7, 5, 131, 19, 17, 11, 37, 5, 7, 13, 5, 7, 11, 5, 23, 5, 67, 7, 5, 41, 29, 13, 11, 7, 5, 17, 5, 19, 7, 5, 43, 13, 7, 5, 31, 37, 5, 11, 67, 7, 5, 7, 17, 5, 11, 5, 7, 23, 5, 37, 7, 19, 59, 5, 11, 13, 47, 5, 7, 5, 11, 7, 5, 13, 5, 7, 5, 7, 5

Offset: 1

N. J. A. Sloane, Feb 28 2004

nonn

It appears that, except for irregular primes belonging to A094095, such as a(5) = 103, a(8) = 37 and a(26) = 59, all regular prime a(n) = p divide the corresponding numerators of the Bernoulli numbers B(A090997(n)) with indices of the form 2*k*p^2, where k > 0 is an integer. - Alexander Adamchuk, Aug 19 2006

The Bernoulli Number Page, Table of factors of the numerators of Bernoulli numbers Bn in the range n = 2..10000, 2018.
S. S. Wagstaff, Jr, Prime factors of the absolute values of Bernoulli numerators, 2018.

Cf. A000367, A090997, A094095, A122270, A122271, A122272, A122273.

In view of the phrase "it appears", it is not clear to me that the correctness of this sequence has been rigorously established. - N. J. A. Sloane, Aug 26 2006
More terms from Alexander Adamchuk, Aug 19 2006
Various sections edited by Petros Hadjicostas, May 12 2020
Corrected and terms a(33) onward added by Max Alekseyev, Mar 16 2023

cp's OEIS Frontend

A090943 Even numbers n such that N(n) is divisible by a nontrivial square, say m^2 with gcd(n,m) = 1, where N(n) is the numerator of the Bernoulli number B(n). The smallest numbers m are given in A094095.

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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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A090997 Numbers m such that the numerator of the Bernoulli number B(m) is divisible by a square.

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A090987 a(n) is the smallest prime whose square divides the numerator of the Bernoulli number B(A090997(n)).

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