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A195441
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a(n) = denominator(Bernoulli_{n+1}(x) - Bernoulli_{n+1}).
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25
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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).
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
The equation a(n-1) = denominator(Bernoulli_n(x) - Bernoulli_n) = rad(n+1) has only finitely many solutions, where rad(n) = A007947(n) is the radical of n. It is conjectured that S = {3, 5, 8, 9, 11, 27, 29, 35, 59} is the full set of all such solutions. Note that (S\{8})+1 joined with {1,2} equals A094960. More precisely, the set S implies the finite sequence of A094960. See Kellner 2023. - Bernd C. Kellner, Oct 18 2023
As was observed in the example section of A318256: denominator(B_n(x)) = rad(n+1) if n is in {0, 1, 3, 5, 9, 11, 27, 29, 35, 59} = {A094960(n) - 1: 1 <= n <= 10}. - Peter Luschny, Oct 18 2023
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LINKS
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FORMULA
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Note that the formulas here are shifted in index by 1 due to the definition of a(n) using index n+1!
a(n-1) = lcm(a(n), rad(n+1)), if n >= 3 is odd.
If n+1 is composite, then rad(n+1) divides a(n-1).
If m is a Carmichael number (A002997), then m divides both a(m-1) and a(m-2).
See papers of Kellner and Kellner & Sondow. (End)
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MAPLE
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A195441 := n -> denom(bernoulli(n+1, x)-bernoulli(n+1)):
# 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 *)
SD[n_, p_] := If[n < 1 || p < 2, 0, Plus @@ IntegerDigits[n, p]]; DD[n_] := Times @@ Select[Prime[Range[PrimePi[(n+2)/(2+Mod[n, 2])]]], SD[n+1, #] >= # &]; Table[DD[n], {n, 0, 59}] (* Bernd C. Kellner, Oct 18 2023 *)
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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])
(Julia)
using Nemo, Primes
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
(Python)
from math import prod
from sympy.ntheory.factor_ import primerange, digits
def A195441(n): return prod(p for p in primerange((n+2)//(2|n&1)+1) if sum(digits(n+1, p)[1:])>=p) # Chai Wah Wu, Oct 04 2023
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CROSSREFS
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Cf. A002997, A064538, A094960, A144845, A286516, A286762, A286763, A318256, A324369, A324370, A324371.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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