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A286516 a(n) = b(2*n-1)/b(2*n) where b(n) = A195441(n-1) = denominator(Bernoulli_{n}(x) - Bernoulli_{n}).

%I A286516 #32 Jul 03 2022 06:47:26
%S A286516 1,2,3,2,5,3,7,2,3,5,11,1,13,7,15,2,17,3,19,5,7,11,23,1,5,13,3,7,29,5,
%T A286516 31,2,11,17,7,1,37,19,13,5,41,21,43,11,3,23,47,1,7,5,17,13,53,3,11,7,
%U A286516 19,29,59,1,61,31,7,2,65,11,67,17,23,5,71,1,73,37
%N A286516 a(n) = b(2*n-1)/b(2*n) where b(n) = A195441(n-1) = denominator(Bernoulli_{n}(x) - Bernoulli_{n}).
%C A286516 a(n) is an integer for all n, a(n) is odd if n is not a power of 2, a(2^k)=2 for all k>=1, a(n)=1 infinitely often, and a(n)=p infinitely often for every prime p. See Cor. 2 and Cor. 3 in "The denominators of power sums of arithmetic progressions". See also "Power-sum denominators".
%H A286516 G. C. Greubel, <a href="/A286516/b286516.txt">Table of n, a(n) for n = 1..1000</a>
%H A286516 Bernd C. Kellner, <a href="https://doi.org/10.1016/j.jnt.2017.03.020">On a product of certain primes</a>, J. Number Theory, 179 (2017), 126-141; arXiv:<a href="https://arxiv.org/abs/1705.04303">1705.04303</a> [math.NT], 2017.
%H A286516 Bernd C. Kellner and Jonathan Sondow, <a href="https://doi.org/10.4169/amer.math.monthly.124.8.695">Power-Sum Denominators</a>, Amer. Math. Monthly, 124 (2017), 695-709; arXiv:<a href="https://arxiv.org/abs/1705.03857">1705.03857</a> [math.NT], 2017.
%H A286516 Bernd C. Kellner and Jonathan Sondow, <a href="http://math.colgate.edu/~integers/s95/s95.pdf">The denominators of power sums of arithmetic progressions</a>, Integers 18 (2018), #A95, 17 pp.; arXiv:<a href="https://arxiv.org/abs/1705.05331">1705.05331</a> [math.NT], 2017.
%F A286516 a(n) = A195441(2*n-2) / A195441(2*n-1).
%t A286516 b[n_] := Denominator[ Together[ BernoulliB[n, x] - BernoulliB[n]]]; Table[
%t A286516 b[2 n - 1]/b[2 n], {n, 1, 74}]
%Y A286516 Cf. A027642, A064538, A144845, A195441, A286515, A286517, A286762, A286763.
%K A286516 nonn
%O A286516 1,2
%A A286516 _Bernd C. Kellner_ and _Jonathan Sondow_, May 12 2017