A366186 Positive integers k such that the third derivative of the k-th Bernoulli polynomial B(k, x) contains only integer coefficients.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 25, 26, 28, 29, 30, 31, 32, 35, 36, 37, 38, 42, 50, 52, 55, 56, 57, 58, 60, 61, 62, 66, 70, 71, 72, 78, 80, 92, 110, 121, 122, 156, 176, 177, 190, 191, 210, 392

Offset: 1

Author

Peter Luschny, Oct 03 2023

Keywords

• nonn
• fini

Comments

From Bernd C. Kellner, Oct 04 2023: (Start)

As a published result on Oct 02 2023 (cf. A366169), all such sequences regarding higher derivatives of the Bernoulli polynomials having only integer coefficients are finite. We have an infinite chain of subsets: A094960 subset of A366169 subset of A366186 subset of A366187 subset of A366188 subset of ... . See Kellner 2023 (Theorem 13, Conjecture 14, and S_3 (this sequence)).

The sequence is finite and is a supersequence of A366169. It remains to show that 392 is the last term. This is very likely, since the terms depend on the estimate 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. (End)

Links

• Olivier Bordellès, Florian Luca, Pieter Moree, and Igor E. Shparlinski, Denominators of Bernoulli polynomials, Mathematika 64 (2018), 519-541.
• Bernd C. Kellner, On a product of certain primes, J. Number Theory, 179 (2017), 126-141; arXiv:1705.04303 [math.NT], 2017.
• Bernd C. Kellner, On the finiteness of Bernoulli polynomials whose derivative has only integral coefficients, 9 pp.; arXiv:2310.01325 [math.NT], 2023.
• Bernd C. Kellner and Jonathan Sondow, Power-Sum Denominators, Amer. Math. Monthly, 124 (2017), 695-709; arXiv:1705.03857 [math.NT], 2017.
• Bernd C. Kellner and Jonathan Sondow, The denominators of power sums of arithmetic progressions, Integers 18 (2018), #A95, 17 pp.; arXiv:1705.05331 [math.NT], 2017.
• Bernd C. Kellner and Jonathan Sondow, On Carmichael and polygonal numbers, Bernoulli polynomials, and sums of base-p digits, Integers 21 (2021), #A52, 21 pp.; arXiv:1902.10672 [math.NT], 2019.

Crossrefs

Cf. A094960 (m=1), A366169 (m=2), this sequence (m=3), A366187 (m=4), A366188 (m=5), A366189.

Programs

Maple

aList := len -> select(n -> denom(diff(bernoulli(n, x), `$`(x, 3))) = 1, [seq(1..len)]): aList(400);

Mathematica

(* From Bernd C. Kellner, Oct 04 2023 (Start) *)
(* k-th derivative of BP *)
k = 3; 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 = 3; Select[Range[1000], DBP[#, k] == 1&]
(* End *)

PARI

isok(k) = #select(x->denominator(x)>1, Vec(deriv(deriv(deriv(bernpol(k)))))) == 0; \\ Michel Marcus, Oct 03 2023

Python

from itertools import count, islice
from sympy import Poly, diff, bernoulli
from sympy.abc import x
def A366186_gen(): # generator of terms
    return filter(lambda k:k<=3 or all(c.is_integer for c in Poly(diff(bernoulli(k,x),x,3)).coeffs()),count(1))
A366186_list = list(islice(A366186_gen(),30)) # Chai Wah Wu, Oct 03 2023

Formula

From Bernd C. Kellner, Oct 04 2023: (Start)

Let (n)_m be the falling factorial. Let s_p(n) be the sum of the p-adic digits of n.

The denominator of the third derivative of the n-th Bernoulli polynomial B(n, x) is given as follows (Kellner 2023, Theorem 12).

D_3(n) = 1 for 1 <= n <= 3. For n > 3, D_3(n) = A324370(n-2)/gcd(A324370(n-2), (n)2) = Product{prime p <= (n-1)/(2+((n-1) mod 2)): gcd(p,(n)_3)=1, s_p(n-2) >= p} p.

Then k is a term if and only if D_3(k) = 1. (End)