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

1,1

Comments

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.

Links

Table of n, a(n) for n=1..17.

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.

Mathematica

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&]

Crossrefs

Cf. A092681, A094095.

Sequence in context: A098245 A343169 A190027 * A252451 A305063 A258550

Adjacent sequences: A090940 A090941 A090942 * A090944 A090945 A090946

Keyword

nonn,nice

Author

T. D. Noe, Feb 27 2004

Extensions

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

Status

approved