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

Author

T. D. Noe, Feb 27 2004

Keywords

• nonn
• nice

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

• 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.

Crossrefs

Cf. A092681, A094095.

Programs

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&]

Extensions

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