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A144845 Least number k such that all coefficients of k*B(n,x), the n-th Bernoulli polynomial, are integers. 18
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 (list; graph; refs; listen; history; text; internal format)
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 condition 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

T. D. Noe, 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.

András Bazsó, István Mező, Some Notes on Alternating Power Sums of Arithmetic Progressions, J. Int. Seq., Vol. 21 (2018), Article 18.7.8.

Bernd C. Kellner and Jonathan Sondow, Power-Sum Denominators, arXiv:1705.03857 [math.NT], 2017; Amer. Math. Monthly, 124 (2017), 695-709.

Bernd C. Kellner and Jonathan Sondow, The denominators of power sums of arithmetic progressions, arXiv:1705.05331 [math.NT], 2017; Integers, 18 (2018), Article A95.

Bernd C. Kellner and Jonathan Sondow, On Carmichael and polygonal numbers, Bernoulli polynomials, and sums of base-p digits, arXiv:1902.10672 [math.NT], 2019.

Eric Weisstein's World of Mathematics, Bernoulli Polynomial

FORMULA

a(n-1) = A324369(n) * A324370(n) * A324371(n) (see Kellner and Sondow 2019).

MAPLE

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

MATHEMATICA

Join[{1}, Table[1/FactorTerms[BernoulliB[n, x], x][[1]], {n, 100}]]

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

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

Sequence in context: A141498 A284004 A225481 * A200563 A284577 A122018

Adjacent sequences:  A144842 A144843 A144844 * A144846 A144847 A144848

KEYWORD

nonn

AUTHOR

T. D. Noe, Sep 22 2008

STATUS

approved

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Last modified April 20 06:31 EDT 2021. Contains 343121 sequences. (Running on oeis4.)