cp's OEIS Frontend

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

%I A366169 #25 May 11 2024 19:24:22
%S A366169 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,
%T A366169 61,70,121,190
%N A366169 Positive integers k such that the second derivative of the k-th Bernoulli polynomial B(k,x) contains only integer coefficients.
%C A366169 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.
%H A366169 Olivier Bordellès, Florian Luca, Pieter Moree, and Igor E. Shparlinski, <a href="https://doi.org/10.1112/S0025579318000153">Denominators of Bernoulli polynomials</a>, Mathematika 64 (2018), 519-541.
%H A366169 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 A366169 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 A366169 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 A366169 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 A366169 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 A366169 k is a term if A366168(k) = 1.
%e A366169 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.
%p A366169 aList := len -> select(n -> denom(diff(diff(bernoulli(n, x), x), x)) = 1, [seq(1..len)]): aList(200);  # _Peter Luschny_, Oct 03 2023
%t A366169 (* k-th derivative of BP *)
%t A366169 k = 2; Select[Range[1000], Denominator[Together[D[BernoulliB[#, x],{x, k}]]] == 1&]
%t A366169 (* exact denominator formula *)
%t A366169 SD[n_, p_] := If[n < 1 || p < 2, 0, Plus@@IntegerDigits[n, p]];
%t A366169 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 A366169 k = 2; Select[Range[1000], DBP[#, k] == 1&]
%o A366169 (PARI) isok(k) = #select(x->denominator(x)>1, Vec(deriv(deriv(bernpol(k))))) == 0; \\ _Michel Marcus_, Oct 03 2023
%o A366169 (Python)
%o A366169 from itertools import count, islice
%o A366169 from sympy import Poly, diff, bernoulli
%o A366169 from sympy.abc import x
%o A366169 def A366169_gen(): # generator of terms
%o A366169     return filter(lambda k:k<=2 or all(c.is_integer for c in Poly(diff(bernoulli(k,x),x,2)).coeffs()),count(1))
%o A366169 A366169_list = list(islice(A366169_gen(),20)) # _Chai Wah Wu_, Oct 03 2023
%Y A366169 Cf. A094960, A144845, A195441, A324370, A366168.
%K A366169 nonn,fini,hard
%O A366169 1,2
%A A366169 _Bernd C. Kellner_, Oct 02 2023