OFFSET
0,2
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.
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), 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) = 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)_m) = A324370(n-m+1)/gcd(A324370(n-m+1), (n)_{m-1}).
(See papers of Kellner and Kellner & Sondow.) (End)
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) *)
PROG
(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
(PARI) a(n) = lcm(apply(x->denominator(x), Vec(bernpol(n)))); \\ Michel Marcus, Mar 03 2020
CROSSREFS
KEYWORD
nonn
AUTHOR
T. D. Noe, Sep 22 2008
STATUS
approved