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a(n) = denominator(Bernoulli_{n}(x)) / denominator(Bernoulli_{n}).
5

%I #47 Jul 03 2022 06:47:32

%S 1,1,1,2,1,6,1,6,1,10,1,6,1,210,5,6,1,30,5,210,7,330,5,30,1,546,7,14,

%T 1,30,1,462,77,3570,35,6,1,51870,455,210,7,2310,55,2310,7,4830,35,210,

%U 1,6630,221,858,11,330,55,798,19,870,5,30,1,930930,5005,4290

%N a(n) = denominator(Bernoulli_{n}(x)) / denominator(Bernoulli_{n}).

%C a(n) is a squarefree integer for all n, a(n) is odd if n>=0 is even, and a(n) is even if n>=3 is odd. See "Power-sum denominators", Thm. 4, pp. 12-13, and "The denominators of power sums of arithmetic progressions", Thm. 3, pp. 3 and 11-12.

%H Robert Israel, <a href="/A286515/b286515.txt">Table of n, a(n) for n = 0..5900</a>

%H 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 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 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 a(n) = A144845(n)/A027642(n) = A195441(n-1)/gcd(A195441(n-1),A027642(n)).

%p seq(denom(bernoulli(n,x))/denom(bernoulli(n)), n=0..100); # _Robert Israel_, May 24 2017

%t Table[ Denominator[ Together[ BernoulliB[n, x]]]/Denominator[ BernoulliB[n]], {n, 0, 63}]

%o (PARI) apply( a(n)=denominator(content(bernpol(n)))/denominator(bernfrac(n)), [1..50]) \\ _M. F. Hasler_, Dec 10 2018

%Y Cf. A027642, A064538, A144845, A195441, A286516, A286517.

%K nonn

%O 0,4

%A _Bernd C. Kellner_ and _Jonathan Sondow_, May 11 2017