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A144845
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Least number k such that all coefficients of k*B(n,x), the n-th Bernoulli polynomial, are integers.
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18
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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
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OFFSET
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0,2
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COMMENTS
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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
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LINKS
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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
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FORMULA
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a(n-1) = A324369(n) * A324370(n) * A324371(n) (see Kellner and Sondow 2019).
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MAPLE
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seq(denom(bernoulli(i, x)), i=0..51); # Peter Luschny, Jun 16 2012
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MATHEMATICA
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Join[{1}, Table[1/FactorTerms[BernoulliB[n, x], x][[1]], {n, 100}]]
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PROG
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(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
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CROSSREFS
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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
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KEYWORD
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nonn
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AUTHOR
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T. D. Noe, Sep 22 2008
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STATUS
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approved
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