|
%I #18 May 12 2020 16:27:41
%S 228,284,914,1434,1616,2948,3292,4280,4336,5612,5768,6302,6944,7714,
%T 7758,8276,9608
%N 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.
%C 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.
%H Bernd Kellner, <a href="http://www.bernoulli.org/~bk/irrpairord.pdf">Über irregulaere Paare hoeherer Ordnungen</a> [On irregular pairs of higher order], Diplomarbeit, Goettingen 2002.
%H S. S. Wagstaff, Jr., <a href="http://www.cerias.purdue.edu/homes/ssw/bernoulli/full.pdf">Prime divisors of the Bernoulli and Euler numbers</a>, 2018.
%H Charles Weibel, <a href="https://dx.doi.org/10.1007/978-3-540-27855-9_5">Algebraic K-Theory of Rings of Integers in Local and Global Fields</a>, 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.
%t 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&]
%Y Cf. A092681, A094095.
%K nonn,nice
%O 1,1
%A _T. D. Noe_, Feb 27 2004
%E Addition of the word "smallest" in the name by _Petros Hadjicostas_, May 12 2020
|
