A144845 Least number k such that all coefficients of k*B(n,x), the n-th Bernoulli polynomial, are integers.

1, 2, 6, 2, 30, 6, 42, 6, 30, 10, 66, 6, 2730, 210, 30, 6, 510, 30, 3990, 210, 2310, 330, 690, 30, 2730, 546, 42, 14, 870, 30, 14322, 462, 39270, 3570, 210, 6, 1919190, 51870, 2730, 210, 94710, 2310, 99330, 2310, 4830, 4830, 9870, 210, 46410, 6630, 14586, 858

Offset: 0

Author

T. D. Noe, Sep 22 2008

Keywords

• nonn

Comments

The lcm of the terms in row n of A053383. It appears that the Bernoulli polynomial B(n,x) is irreducible for all even n.

This sequence appears in a paper of Bazsó & Mező, who use this sequence to give necessary and sufficient conditions for power sums to be integer polynomials. - Istvan Mezo, Mar 20 2016

In "The denominators of power sums of arithmetic progressions" Corollary 1, we give a simple way to compute a(n) without using Bernoulli polynomials. Namely, a(n) equals (product of the primes dividing n+1) times (product of the primes p <= (n+1)/(2+(n+1 mod 2)) not dividing n+1 such that the sum of the base-p digits of n+1 is at least p). - Bernd C. Kellner and Jonathan Sondow, May 15 2017

Links

• Bernd C. Kellner, Table of n, a(n) for n = 0..10000 (n = 0..1000 from T. D. Noe)
• András Bazsó and István Mező, On the coefficients of power sums of arithmetic progressions, J. Number Th., 153 (2015), 117-123.
• András Bazsó and István Mező, Some Notes on Alternating Power Sums of Arithmetic Progressions, J. Int. Seq., Vol. 21 (2018), Article 18.7.8.
• 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, On the finiteness of Bernoulli polynomials whose derivative has only integral coefficients, J. Integer Seq. 27 (2024), Article 24.2.8, 11 pp.; arXiv:2310.01325 [math.NT], 2023.
• 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.
• Eric Weisstein's World of Mathematics, Bernoulli Polynomial.

Crossrefs

Cf. A027642, A053383, A064538, A195441, A286515, A286516, A286517, A324369, A324370, A324371.

Programs

Maple

seq(denom(bernoulli(i,x)),i=0..51); # Peter Luschny, Jun 16 2012

Mathematica

(* From Bernd C. Kellner, Oct 18 2023: (Start) *)
(* Denominator formula *)
Table[Denominator[Together[BernoulliB[n, x]]], {n, 0, 51}]
(* Product formula *)
SD[n_, p_] := If[n < 1 || p < 2, 0, Plus@@IntegerDigits[n, p]]; rad[n_] := Times @@ Select[Divisors[n], PrimeQ]; (* A324370 *) DD2[n_] := Times @@ Select[Prime[Range[PrimePi[(n+1)/(2+Mod[n+1, 2])]]], !Divisible[n, #] && SD[n, #] >= # &];
DB[n_] := DD2[n+1] rad[n+1]; Table[DB[n], {n, 0, 51}]
(* (End) *)

PARI

a(n) = lcm(apply(x->denominator(x), Vec(bernpol(n)))); \\ Michel Marcus, Mar 03 2020

Sage

def A144845(n):
    return mul(prime_divisors(n+1) + [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([A144845(n) for n in (0..51)]) # Peter Luschny, Sep 12 2018

Formula

From Bernd C. Kellner, Oct 18 2023: (Start)

Let rad(n) = A007947(n) be the radical of n. Let (n)_m be the falling factorial. Let f^(m)(x) denote the m-th derivative of f(x).

a(n) = lcm(A195441(n-1), A027642(n)) = lcm(denom(B(n,x)-B_n), denom(B_n)) = denom(B(n,x)).

a(n) = lcm(A195441(n), rad(n+1)).

a(n) = lcm(a(n+1), rad(n+1)), if n >= 2 is even.

a(2n)/a(2n+1) = A286517(n), if n >= 1.

a(n) = A324369(n+1) * A324370(n+1) * A324371(n+1).

a(n) = A324370(n+1) * rad(n+1).

a(n) = rad(A064538(n)).

If n >= m >= 1, then denom(B^(m)(n,x)) = a(n-m)/gcd(a(n-m), (n)A324370(n-m+1)/gcd(A324370(n-m+1),%20(n)">m) = A324370(n-m+1)/gcd(A324370(n-m+1), (n){m-1}).

(See papers of Kellner and Kellner & Sondow.) (End)