A366169 Positive integers k such that the second derivative of the k-th Bernoulli polynomial B(k,x) contains only integer coefficients.

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 15, 16, 21, 25, 28, 29, 30, 31, 36, 37, 55, 57, 60, 61, 70, 121, 190

Offset: 1

Author

Bernd C. Kellner, Oct 02 2023

Keywords

• nonn
• fini
• hard

Comments

The sequence is finite and is a supersequence of A094960. The terms are those numbers k where the denominator A366168(k) = 1. It remains to show that 190 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.

Examples

			B(5,x) = x^5 - (5x^4)/2 + (5 x^3)/3 - x/6 and B''(5,x) = 20x^3 - 30x^2 + 10x, so 5 is a term.
		

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, J. Integer Seq. 27 (2024), Article 24.2.8, 11 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, A144845, A195441, A324370, A366168.

Programs

Maple

aList := len -> select(n -> denom(diff(diff(bernoulli(n, x), x), x)) = 1, [seq(1..len)]): aList(200);  # Peter Luschny, Oct 03 2023

Mathematica

(* k-th derivative of BP *)
k = 2; 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 = 2; Select[Range[1000], DBP[#, k] == 1&]

PARI

isok(k) = #select(x->denominator(x)>1, Vec(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 A366169_gen(): # generator of terms
    return filter(lambda k:k<=2 or all(c.is_integer for c in Poly(diff(bernoulli(k,x),x,2)).coeffs()),count(1))
A366169_list = list(islice(A366169_gen(),20)) # Chai Wah Wu, Oct 03 2023

Formula

k is a term if A366168(k) = 1.