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A195441
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a(n) = denominator(Bernoulli_{n+1}(x) - Bernoulli_{n+1}).
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22
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1, 1, 2, 1, 6, 2, 6, 3, 10, 2, 6, 2, 210, 30, 6, 3, 30, 10, 210, 42, 330, 30, 30, 30, 546, 42, 14, 2, 30, 2, 462, 231, 3570, 210, 6, 2, 51870, 2730, 210, 42, 2310, 330, 2310, 210, 4830, 210, 210, 210, 6630, 1326, 858, 66, 330, 110, 798, 114, 870, 30, 30, 6
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OFFSET
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0,3
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COMMENTS
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If s(n) is the smallest number such that s(n)*(1^n + 2^n + … + x^n) is a polynomial in x with integer coefficients then a(n)=s(n)/(n+1) (see A064538).
a(n) is squarefree, by the von Staudt-Clausen theorem on the denominators of Bernoulli numbers. - Kieren MacMillan and Jonathan Sondow, Nov 20 2015
Kellner and Sondow give a detailed analysis of this sequence and provide a simple way to compute the terms without using Bernoulli polynomials and numbers. They prove that a(n) is the product of the primes less than or equal to (n+2)/(2+(n mod 2)) such that the sum of digits of n+1 in base p is at least p. - Peter Luschny, May 14 2017
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LINKS
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Peter Luschny, Table of n, a(n) for n = 0..10000 (terms 0..1000 from G. C. Greubel)
Harald Hofstätter, Denominators of coefficients of the Baker-Campbell-Hausdorff series, arXiv:2010.03440 [math.NT], 2020. Mentions this sequence.
Bernd C. Kellner, On a product of certain primes, arXiv:1705.04303 [math.NT] 2017; J. Number Theory, 179 (2017), 126-141.
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.
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FORMULA
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a(n) = A064538(n)/(n+1). - Jonathan Sondow, Nov 12 2015
A001221(a(n)) = A001222(a(n)). - Kieren MacMillan and Jonathan Sondow, Nov 20 2015
a(2*n)/a(2*n+1) = A286516(n+1). - Bernd C. Kellner and Jonathan Sondow, May 24 2017
a(n) = A007947(A338025(n+1)). - Harald Hofstätter, Oct 10 2020
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MAPLE
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A195441 := n -> denom(bernoulli(n+1, x)-bernoulli(n+1)):
seq(A195441(i), i=0..59);
# Formula of Kellner and Sondow:
a := proc(n) local s; s := (p, n) -> add(i, i=convert(n, base, p));
select(isprime, [$2..(n+2)/(2+irem(n, 2))]); mul(i, i=select(p->s(p, n+1)>=p, %)) end: seq(a(n), n=0..59); # Peter Luschny, May 14 2017
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MATHEMATICA
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a[n_] := Denominator[ Together[(BernoulliB[n + 1, x] - BernoulliB[n + 1])]]; Table[a[n], {n, 0, 59}] (* Jonathan Sondow, Nov 20 2015 *)
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PROG
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(PARI) a(n) = {my(vp = Vec(bernpol(n+1, x)-bernfrac(n+1))); lcm(vector(#vp, k, denominator(vp[k]))); } \\ Michel Marcus, Feb 08 2016
(Sage)
A195441 = lambda n: mul([p for p in (2..(n+2)//(2+n%2)) if is_prime(p) and sum((n+1).digits(base=p))>=p])
print([A195441(n) for n in (0..59)]) # Peter Luschny, May 14 2017
(Julia)
using Nemo, Primes
function A195441(n::Int)
n < 4 && return ZZ([1, 1, 2, 1][n+1])
P = primes(2, div(n+2, 2+n%2))
prod([ZZ(p) for p in P if p <= sum(digits(n+1, base=p))])
end
println([A195441(n) for n in 0:59]) # Peter Luschny, May 14 2017
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CROSSREFS
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Cf. A064538, A286516, A286762, A286763.
Sequence in context: A306549 A198870 A050457 * A338025 A239537 A076891
Adjacent sequences: A195438 A195439 A195440 * A195442 A195443 A195444
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KEYWORD
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nonn
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AUTHOR
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Peter Luschny, Sep 18 2011
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EXTENSIONS
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Definition simplified by Jonathan Sondow, Nov 20 2015
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STATUS
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approved
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