A286515 a(n) = denominator(Bernoulli_{n}(x)) / denominator(Bernoulli_{n}).

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, 1, 30, 1, 462, 77, 3570, 35, 6, 1, 51870, 455, 210, 7, 2310, 55, 2310, 7, 4830, 35, 210, 1, 6630, 221, 858, 11, 330, 55, 798, 19, 870, 5, 30, 1, 930930, 5005, 4290

Offset: 0

Author

Bernd C. Kellner and Jonathan Sondow, May 11 2017

Keywords

• nonn

Comments

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.

Links

• Robert Israel, Table of n, a(n) for n = 0..5900
• Bernd C. Kellner, On a product of certain primes, J. Number Theory, 179 (2017), 126-141; arXiv:1705.04303 [math.NT], 2017.
• Bernd C. Kellner and Jonathan Sondow, Power-Sum Denominators, Amer. Math. Monthly, 124 (2017), 695-709; arXiv:1705.03857 [math.NT], 2017.
• Bernd C. Kellner and Jonathan Sondow, The denominators of power sums of arithmetic progressions, Integers 18 (2018), #A95, 17 pp.; arXiv:1705.05331 [math.NT], 2017.

Crossrefs

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

Programs

Maple

seq(denom(bernoulli(n,x))/denom(bernoulli(n)), n=0..100); # Robert Israel, May 24 2017

Mathematica

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

PARI

apply( a(n)=denominator(content(bernpol(n)))/denominator(bernfrac(n)), [1..50]) \\ M. F. Hasler, Dec 10 2018

Formula

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