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%I #52 Oct 04 2023 19:03:47
%S 1,2,4,6,10,12,28,30,36,60
%N Positive integers k such that the derivative of the k-th Bernoulli polynomial B(k,x) contains only integer coefficients.
%C From _Max Alekseyev_, Dec 08 2011: (Start)
%C There are no other terms below 10^9.
%C k belongs to this sequence if k*binomial(k-1,m)*Bernoulli(m) is an integer for each m in 0..k-1. (End)
%C From _Max Alekseyev_, Jun 04 2012: (Start)
%C If for a prime p >= 3, k ends with base-p digits a,b with a+b >= p, then for m = (a+1)*(p-1), the number k*binomial(k-1,m)*Bernoulli(m) is not an integer (it contains p in the denominator). For p=3, this implies that k == 5, 7, or 8 (mod 9) are not in this sequence; for p=5, this implies that k == 9, 13, 14, 17, 18, 19, 21, 22, 23, or 24 (mod 25) are not in this sequence; and so on.
%C Conjecture: for every integer k > 78, there exists a prime p >= 3 such that the sum of last two base-p digits of k is at least p. This conjecture would imply that this sequence is finite and 60 is the last term. (End)
%C The conjecture is true for all k such that k+1 is not a prime, a power of 2, or a Giuga number (A007850). In this case, there exists a prime p >= 3 such that the base-p representation of k ends in a,p-1 with a > 0. - _Max Alekseyev_, Feb 16 2021
%C The sequence is finite and is a subsequence of A366169. The terms are those numbers k where A324370(k) = 1. It remains to show that 60 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. - _Bernd C. Kellner_, Oct 02 2023
%H 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 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 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 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 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 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 k is a term if A324370(k) = 1. - _Bernd C. Kellner_, Oct 02 2023
%F k is a term <=> 0 = Sum_{j=0..k-1} k*binomial(k - 1, j) mod Clausen(j), where Clausen(n) = A160014(n, 1). - _Peter Luschny_, Oct 04 2023
%e B(6,x) = x^6 - 3*x^5 + (5/2)*x^4 - (1/2)*x^2 + 1/42 so B'(6,x) contains only integer coefficients and 6 is in the sequence.
%p p := n -> if denom(diff(bernoulli(n, x), x)) = 1 then n else fi:
%p seq(p(n), n=1..100); # _Emeric Deutsch_
%t (* From _Bernd C. Kellner_, Oct 02 2023. (Start) *)
%t (* k-th derivative of BP: *)
%t k = 1; Select[Range[1000], Denominator[Together[D[BernoulliB[#, x],{x, k}]]] == 1&]
%t (* Exact denominator formula: *)
%t SD[n_, p_] := If[n < 1 || p < 2, 0, Plus@@IntegerDigits[n, p]];
%t 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 k = 1; Select[Range[1000], DBP[#, k] == 1&]
%t (* End *)
%o (PARI) is_A094960(k) = !#select(x->(denominator(x)!=1), Vec(deriv(bernpol(k)))); \\ _Michel Marcus_, Feb 15 2021
%o (Python)
%o from itertools import count, islice
%o from sympy import Poly, diff, bernoulli
%o from sympy.abc import x
%o def A094960_gen(): # generator of terms
%o return filter(lambda k:k<=1 or all(c.is_integer for c in Poly(diff(bernoulli(k,x),x)).coeffs()),count(1))
%o A094960_list = list(islice(A094960_gen(),10)) # _Chai Wah Wu_, Oct 03 2023
%Y Cf. A094960, A144845, A160014, A195441, A324370, A366168, A366169.
%K nonn,fini,hard
%O 1,2
%A _Benoit Cloitre_, Jun 19 2004
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