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A094960
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Positive integers k such that the derivative of the k-th Bernoulli polynomial B(k,x) contains only integer coefficients.
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9
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OFFSET
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1,2
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COMMENTS
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There are no other terms below 10^9.
k belongs to this sequence if k*binomial(k-1,m)*Bernoulli(m) is an integer for each m in 0..k-1. (End)
If for a prime p >= 3, k ends with base-p digits a,b with a+b >= p, then for m = (a+1)*(p-1), the number k*binomial(k-1,m)*Bernoulli(m) is not an integer (it contains p in the denominator). For p=3, this implies that k == 5, 7, or 8 (mod 9) are not in this sequence; for p=5, this implies that k == 9, 13, 14, 17, 18, 19, 21, 22, 23, or 24 (mod 25) are not in this sequence; and so on.
Conjecture: for every integer k > 78, there exists a prime p >= 3 such that the sum of last two base-p digits of k is at least p. This conjecture would imply that this sequence is finite and 60 is the last term. (End)
The conjecture is true for all k such that k+1 is not a prime, a power of 2, or a Giuga number (A007850). In this case, there exists a prime p >= 3 such that the base-p representation of k ends in a,p-1 with a > 0. - Max Alekseyev, Feb 16 2021
The sequence is finite and is a subsequence of A366169. The terms are those numbers k where A324370(k) = 1. It remains to show that 60 is the last term. This is very likely, since the terms depend on the estimation of a product of primes satisfying certain p-adic conditions that is connected with A324370. A proven asymptotic formula related to that product implies that this sequence is finite. See Kellner 2017, 2023, and BLMS 2018. - Bernd C. Kellner, Oct 02 2023
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LINKS
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FORMULA
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k is a term <=> 0 = Sum_{j=0..k-1} k*binomial(k - 1, j) mod Clausen(j), where Clausen(n) = A160014(n, 1). - Peter Luschny, Oct 04 2023
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EXAMPLE
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B(6,x) = x^6 - 3*x^5 + (5/2)*x^4 - (1/2)*x^2 + 1/42 so B'(6,x) contains only integer coefficients and 6 is in the sequence.
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MAPLE
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p := n -> if denom(diff(bernoulli(n, x), x)) = 1 then n else fi:
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MATHEMATICA
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(* k-th derivative of BP: *)
k = 1; Select[Range[1000], Denominator[Together[D[BernoulliB[#, x], {x, k}]]] == 1&]
(* Exact denominator formula: *)
SD[n_, p_] := If[n < 1 || p < 2, 0, Plus@@IntegerDigits[n, p]];
DBP[n_, k_] := Module[{m = n-k+1, fac = FactorialPower[n, k]}, If[n < 1 || k < 1 || n <= k, Return[1]]; Times@@Select[Prime[Range[PrimePi[(m+1)/(2 + Mod[m+1, 2])]]], !Divisible[fac, #] && SD[m, #] >= #&]];
k = 1; Select[Range[1000], DBP[#, k] == 1&]
(* End *)
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PROG
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(PARI) is_A094960(k) = !#select(x->(denominator(x)!=1), Vec(deriv(bernpol(k)))); \\ Michel Marcus, Feb 15 2021
(Python)
from itertools import count, islice
from sympy import Poly, diff, bernoulli
from sympy.abc import x
def A094960_gen(): # generator of terms
return filter(lambda k:k<=1 or all(c.is_integer for c in Poly(diff(bernoulli(k, x), x)).coeffs()), count(1))
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CROSSREFS
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KEYWORD
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nonn,fini,hard
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AUTHOR
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STATUS
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approved
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