A366168 Denominator of the second derivative of the n-th Bernoulli polynomial B(n,x).

1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 15, 1, 1, 3, 5, 5, 21, 1, 5, 15, 5, 1, 21, 7, 1, 1, 1, 1, 231, 7, 35, 3, 1, 1, 1365, 35, 7, 21, 55, 55, 105, 7, 7, 105, 35, 5, 663, 13, 11, 33, 55, 1, 57, 1, 5, 15, 1, 1, 15015, 715, 715, 33, 17, 85, 2415, 35, 1, 3, 55, 55, 285285, 19019, 1001

Offset

1,8

Comments

The sequence consists only of odd numbers. The denominators are connected with A324370, from which an explicit formula follows as given below. See Kellner 2023.

Links

Bernd C. Kellner, Table of n, a(n) for n = 1..10000

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.

Formula

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

a(1) = 1, and for n > 1, a(n) = A324370(n-1)/gcd(A324370(n-1), n) = Product_{prime p <= n/(2+(n mod 2)): gcd(p,(n)_2)=1, s_p(n-1) >= p} p.

Example

B(5,x) = x^5 - (5x^4)/2 + (5 x^3)/3 - x/6 and B''(5,x) = 20x^3 - 30x^2 + 10x, so a(5) = 1.
a(14) = A324370(13)/gcd(A324370(13), 14) = 210/gcd(210, 14) = 15.

Mathematica

(* k-th derivative of BP *)
k = 2; Table[Denominator[Together[D[BernoulliB[n, x], {x, k}]]], {n, 1, 100}]
(* 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; Table[DBP[n, k], {n, 1, 100}]

Prog

(Python)
from math import lcm
from sympy import Poly, diff, bernoulli
from sympy.abc import x
def A366168(n): return lcm(*(c.q for c in Poly(diff(bernoulli(n, x), x, 2)).coeffs())) if n>=3 else 1 # Chai Wah Wu, Oct 04 2023

Crossrefs

Cf. A094960, A144845, A195441, A324370, A366169.

Sequence in context: A061653 A069226 A326697 * A016565 A051714 A023593

Adjacent sequences: A366165 A366166 A366167 * A366169 A366170 A366171

Keyword

nonn,frac

Author

Bernd C. Kellner, Oct 02 2023

Status

approved