A318256 a(n) = (denominator of B(n,x)) / (the squarefree kernel of n+1), where B(n,x) is the n-th Bernoulli polynomial.

1, 1, 2, 1, 6, 1, 6, 3, 10, 1, 6, 1, 210, 15, 2, 3, 30, 5, 210, 21, 110, 15, 30, 5, 546, 21, 14, 1, 30, 1, 462, 231, 1190, 105, 6, 1, 51870, 1365, 70, 21, 2310, 55, 2310, 105, 322, 105, 210, 35, 6630, 663, 286, 33, 330, 55, 798, 57, 290, 15, 30, 1, 930930, 15015

Offset: 0

Author

Peter Luschny, Sep 12 2018

Keywords

• nonn

Examples

			a(59) = 1 because there exist no number which satisfies the definition (and the product of an empty set is 1).
a(60) = 930930 because {2, 3, 5, 7, 11, 13, 31} are the only primes which satisfy the definition.
The denominator of the Bernoulli polynomial B_n(x) equals the squarefree kernel of n+1 if n is in {0, 1, 3, 5, 9, 11, 27, 29, 35, 59}. These might be the only numbers with this property.
		

Links

• Peter Luschny, Table of n, a(n) for n = 0..1000
• András Bazsó and István Mező, On the coefficients of power sums of arithmetic progressions, J. Number Th., 153 (2015), 117-123.
• 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.
• Bernd C. Kellner and Jonathan Sondow, On Carmichael and polygonal numbers, Bernoulli polynomials, and sums of base-p digits, Integers 21 (2021), #A52, 21 pp.; arXiv:1902.10672 [math.NT], 2019.
• Bernd C. Kellner, On the finiteness of Bernoulli polynomials whose derivative has only integral coefficients, 9 pp.; arXiv:2310.01325 [math.NT], 2023.

Crossrefs

a(n) = A144845(n) / A007947(n+1).

Cf. A324370 (same sequence with offset 1).

Cf. A027642, A064538, A195441, A286515, A286516, A286517, A286762, A286763, A319084.

Programs

Maple

a := n -> denom(bernoulli(n, x)) / mul(p, p in numtheory:-factorset(n+1)):
seq(a(n), n=0..61);

Mathematica

sfk[n_] := Times @@ FactorInteger[n][[All, 1]];
a[n_] := (BernoulliB[n, x] // Together // Denominator)/sfk[n+1];
Table[a[n], {n, 0, 61}] (* Jean-François Alcover, Feb 14 2019 *)

Sage

def A318256(n): return mul([p for p in (2..(n+2)//(2+n%2))
                if is_prime(p)
                and not p.divides(n+1)
                and sum((n+1).digits(base=p)) >= p])
print([A318256(n) for n in (0..61)])

Formula

Let Q(n) = {p <= floor((n + 2)/(2 + n mod 2)) and p is prime and p does not divide n + 1 and the sum of the digits in base p of n+1 is at least p} then a(n) = Product_{p in Q(n)} p. (See the Kellner & Sondow links.)

a(n) = denominator(Bernoulli'(n+1, x)), where ' denotes d/dx. - Peter Luschny, Oct 15 2023