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A366186 Positive integers k such that the third derivative of the k-th Bernoulli polynomial B(k, x) contains only integer coefficients.

%I A366186 #22 Oct 08 2023 04:51:54
%S A366186 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,20,21,22,25,26,28,29,30,
%T A366186 31,32,35,36,37,38,42,50,52,55,56,57,58,60,61,62,66,70,71,72,78,80,92,
%U A366186 110,121,122,156,176,177,190,191,210,392
%N A366186 Positive integers k such that the third derivative of the k-th Bernoulli polynomial B(k, x) contains only integer coefficients.
%C A366186 From _Bernd C. Kellner_, Oct 04 2023: (Start)
%C A366186 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, Conjecture 14, and S_3 (this sequence)).
%C A366186 The sequence is finite and is a supersequence of A366169. It remains to show that 392 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 A366186 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 A366186 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 A366186 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 A366186 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 A366186 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 A366186 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 A366186 From _Bernd C. Kellner_, Oct 04 2023: (Start)
%F A366186 Let (n)_m be the falling factorial. Let s_p(n) be the sum of the p-adic digits of n.
%F A366186 The denominator of the third derivative of the n-th Bernoulli polynomial B(n, x) is given as follows (Kellner 2023, Theorem 12).
%F A366186 D_3(n) = 1 for 1 <= n <= 3. For n > 3, D_3(n) = A324370(n-2)/gcd(A324370(n-2), (n)_2) = Product_{prime p <= (n-1)/(2+((n-1) mod 2)): gcd(p,(n)_3)=1, s_p(n-2) >= p} p.
%F A366186 Then k is a term if and only if D_3(k) = 1. (End)
%p A366186 aList := len -> select(n -> denom(diff(bernoulli(n, x), `$`(x, 3))) = 1, [seq(1..len)]): aList(400);
%t A366186 (* From _Bernd C. Kellner_, Oct 04 2023 (Start) *)
%t A366186 (* k-th derivative of BP *)
%t A366186 k = 3; Select[Range[1000], Denominator[Together[D[BernoulliB[#, x], {x, k}]]] == 1&]
%t A366186 (* Exact denominator formula *)
%t A366186 SD[n_, p_] := If[n < 1 || p < 2, 0, Plus@@IntegerDigits[n, p]];
%t A366186 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 A366186 k = 3; Select[Range[1000], DBP[#, k] == 1&]
%t A366186 (* End *)
%o A366186 (PARI) isok(k) = #select(x->denominator(x)>1, Vec(deriv(deriv(deriv(bernpol(k)))))) == 0; \\ _Michel Marcus_, Oct 03 2023
%o A366186 (Python)
%o A366186 from itertools import count, islice
%o A366186 from sympy import Poly, diff, bernoulli
%o A366186 from sympy.abc import x
%o A366186 def A366186_gen(): # generator of terms
%o A366186     return filter(lambda k:k<=3 or all(c.is_integer for c in Poly(diff(bernoulli(k,x),x,3)).coeffs()),count(1))
%o A366186 A366186_list = list(islice(A366186_gen(),30)) # _Chai Wah Wu_, Oct 03 2023
%Y A366186 Cf. A094960 (m=1), A366169 (m=2), this sequence (m=3), A366187 (m=4), A366188 (m=5), A366189.
%K A366186 nonn,fini
%O A366186 1,2
%A A366186 _Peter Luschny_, Oct 03 2023