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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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28
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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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FORMULA
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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(a(n+1), rad(n+1)), if n >= 2 is even.
a(2n)/a(2n+1) = A286517(n), if n >= 1.
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)
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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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(* 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) *)
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PROG
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(Sage)
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])
(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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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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