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

%I A366168 #22 May 11 2024 19:59:25
%S A366168 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,
%T A366168 231,7,35,3,1,1,1365,35,7,21,55,55,105,7,7,105,35,5,663,13,11,33,55,1,
%U A366168 57,1,5,15,1,1,15015,715,715,33,17,85,2415,35,1,3,55,55,285285,19019,1001
%N A366168 Denominator of the second derivative of the n-th Bernoulli polynomial B(n,x).
%C A366168 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.
%H A366168 Bernd C. Kellner, <a href="/A366168/b366168.txt">Table of n, a(n) for n = 1..10000</a>
%H A366168 Bernd C. Kellner, <a href="https://doi.org/10.1016/j.jnt.2017.03.020">On a product of certain primes</a>, J. Number Theory, 179 (2017), 126-141; arXiv:<a href="https://arxiv.org/abs/1705.04303">1705.04303</a> [math.NT], 2017.
%H A366168 Bernd C. Kellner, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL27/Kellner/kell2.html">On the finiteness of Bernoulli polynomials whose derivative has only integral coefficients</a>, J. Integer Seq. 27 (2024), Article 24.2.8, 11 pp.; arXiv:<a href="https://arxiv.org/abs/2310.01325">2310.01325</a> [math.NT], 2023.
%H A366168 Bernd C. Kellner and Jonathan Sondow, <a href="https://doi.org/10.4169/amer.math.monthly.124.8.695">Power-Sum Denominators</a>, Amer. Math. Monthly, 124 (2017), 695-709; arXiv:<a href="https://arxiv.org/abs/1705.03857">1705.03857</a> [math.NT], 2017.
%H A366168 Bernd C. Kellner and Jonathan Sondow, <a href="http://math.colgate.edu/~integers/s95/s95.pdf">The denominators of power sums of arithmetic progressions</a>, Integers 18 (2018), #A95, 17 pp.; arXiv:<a href="https://arxiv.org/abs/1705.05331">1705.05331</a> [math.NT], 2017.
%H A366168 Bernd C. Kellner and Jonathan Sondow, <a href="http://math.colgate.edu/~integers/v52/v52.pdf">On Carmichael and polygonal numbers, Bernoulli polynomials, and sums of base-p digits</a>, Integers 21 (2021), #A52, 21 pp.; arXiv:<a href="https://arxiv.org/abs/1902.10672">1902.10672</a> [math.NT], 2019.
%F A366168 Let (n)_k be the falling factorial. Let s_p(n) be the sum of the p-adic digits of n.
%F A366168 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.
%e A366168 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.
%e A366168 a(14) = A324370(13)/gcd(A324370(13), 14) = 210/gcd(210, 14) = 15.
%t A366168 (* k-th derivative of BP *)
%t A366168 k = 2; Table[Denominator[Together[D[BernoulliB[n, x], {x, k}]]], {n, 1, 100}]
%t A366168 (* exact denominator formula *)
%t A366168 SD[n_, p_] := If[n < 1 || p < 2, 0, Plus@@IntegerDigits[n, p]];
%t A366168 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, #] >= #&]];
%t A366168 k = 2; Table[DBP[n, k], {n, 1, 100}]
%o A366168 (Python)
%o A366168 from math import lcm
%o A366168 from sympy import Poly, diff, bernoulli
%o A366168 from sympy.abc import x
%o A366168 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
%Y A366168 Cf. A094960, A144845, A195441, A324370, A366169.
%K A366168 nonn,frac
%O A366168 1,8
%A A366168 _Bernd C. Kellner_, Oct 02 2023