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

0,3

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.

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

Example

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.

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 *)

Prog

(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)])

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.

Sequence in context: A366570 A286515 A166120 * A324370 A324193 A364829

Adjacent sequences: A318253 A318254 A318255 * A318257 A318258 A318259

Keyword

nonn

Author

Peter Luschny, Sep 12 2018

Status

approved