

A094960


Positive integers k such that the derivative of the kth Bernoulli polynomial B(k,x) contains only integer coefficients.


9




OFFSET

1,2


COMMENTS

There are no other terms below 10^9.
k belongs to this sequence if k*binomial(k1,m)*Bernoulli(m) is an integer for each m in 0..k1. (End)
If for a prime p >= 3, k ends with basep digits a,b with a+b >= p, then for m = (a+1)*(p1), the number k*binomial(k1,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 basep 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 basep representation of k ends in a,p1 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 padic 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


LINKS



FORMULA

k is a term <=> 0 = Sum_{j=0..k1} k*binomial(k  1, j) mod Clausen(j), where Clausen(n) = A160014(n, 1).  Peter Luschny, Oct 04 2023


EXAMPLE

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.


MAPLE

p := n > if denom(diff(bernoulli(n, x), x)) = 1 then n else fi:


MATHEMATICA

(* kth 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 = nk+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 *)


PROG

(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))


CROSSREFS



KEYWORD

nonn,fini,hard


AUTHOR



STATUS

approved



