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A366187 Positive integers k such that the fourth derivative of the k-th Bernoulli polynomial B(k, x) contains only integer coefficients.

%I A366187 #20 Oct 08 2023 04:51:24
%S A366187 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,25,26,27,
%T A366187 28,29,30,31,32,33,35,36,37,38,39,42,43,45,50,51,52,53,55,56,57,58,59,
%U A366187 60,61,62,63,66,67,68,70,71,72,73,78,79,80,81,91,92,93,110,111,121,122,123,143,147,156,157,171,176,177,178,190,191,192,210,211,255,392,393
%N A366187 Positive integers k such that the fourth derivative of the k-th Bernoulli polynomial B(k, x) contains only integer coefficients.
%C A366187 From _Bernd C. Kellner_, Oct 04 2023: (Start)
%C A366187 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).
%C A366187 The sequence is finite and is a supersequence of A366186. It remains to show that 393 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)
%H A366187 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 A366187 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 A366187 Bernd C. Kellner, <a href="https://arxiv.org/abs/2310.01325">On the finiteness of Bernoulli polynomials whose derivative has only integral coefficients</a>, 9 pp.; arXiv:2310.01325 [math.NT], 2023.
%H A366187 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 A366187 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 A366187 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 A366187 From _Bernd C. Kellner_, Oct 04 2023: (Start)
%F A366187 Let (n)_m be the falling factorial. Let s_p(n) be the sum of the p-adic digits of n.
%F A366187 The denominator of the fourth derivative of the n-th Bernoulli polynomial B(n, x) is given as follows (Kellner 2023, Theorem 12).
%F A366187 D_4(n) = 1 for 1 <= n <= 4. For n > 4, D_4(n) = A324370(n-3)/gcd(A324370(n-3), (n)_3) = Product_{prime p <= (n-2)/(2+((n-2) mod 2)): gcd(p,(n)_4)=1, s_p(n-3) >= p} p.
%F A366187 Then k is a term if and only if D_4(k) = 1. (End)
%p A366187 aList := len -> select(n -> denom(diff(bernoulli(n, x), `$`(x, 4))) = 1, [seq(1..len)]): aList(400);
%t A366187 (* From _Bernd C. Kellner_, Oct 04 2023 (Start) *)
%t A366187 (* k-th derivative of BP *)
%t A366187 k = 4; Select[Range[1000], Denominator[Together[D[BernoulliB[#, x], {x, k}]]] == 1&]
%t A366187 (* Exact denominator formula *)
%t A366187 SD[n_, p_] := If[n < 1 || p < 2, 0, Plus@@IntegerDigits[n, p]];
%t A366187 DBP[n_,
%t A366187  k_] := Module[{m = n-k+1, fac = FactorialPower[n, k]}, If[n < 1 || k
%t A366187  < 1 || n <= k, Return[1]];
%t A366187 Times@@Select[Prime[Range[PrimePi[(m+1)/(2 + Mod[m+1, 2])]]],
%t A366187 !Divisible[fac, #] && SD[m, #] >= #&]];
%t A366187 k = 4; Select[Range[1000], DBP[#, k] == 1&]
%t A366187 (* End *)
%o A366187 (PARI) isok(k) = #select(x->denominator(x)>1, Vec(deriv(deriv(deriv(deriv(bernpol(k))))))) == 0; \\ _Michel Marcus_, Oct 03 2023
%o A366187 (Python)
%o A366187 from itertools import count, islice
%o A366187 from sympy import Poly, diff, bernoulli
%o A366187 from sympy.abc import x
%o A366187 def A366187_gen(): # generator of terms
%o A366187     return filter(lambda k:k<=4 or all(c.is_integer for c in Poly(diff(bernoulli(k,x),x,4)).coeffs()),count(1))
%o A366187 A366187_list = list(islice(A366187_gen(),40)) # _Chai Wah Wu_, Oct 03 2023
%Y A366187 Cf. A094960 (m=1), A366169 (m=2), A366186 (m=3), this sequence (m=4), A366188 (m=5), A366189.
%K A366187 nonn,fini
%O A366187 1,2
%A A366187 _Peter Luschny_, Oct 03 2023