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A318256 a(n) = (denominator of B(n,x)) / (the squarefree kernel of n+1), where B(n,x) is the n-th Bernoulli polynomial.

%I A318256 #37 Oct 15 2023 09:25:43
%S A318256 1,1,2,1,6,1,6,3,10,1,6,1,210,15,2,3,30,5,210,21,110,15,30,5,546,21,
%T A318256 14,1,30,1,462,231,1190,105,6,1,51870,1365,70,21,2310,55,2310,105,322,
%U A318256 105,210,35,6630,663,286,33,330,55,798,57,290,15,30,1,930930,15015
%N A318256 a(n) = (denominator of B(n,x)) / (the squarefree kernel of n+1), where B(n,x) is the n-th Bernoulli polynomial.
%H A318256 Peter Luschny, <a href="/A318256/b318256.txt">Table of n, a(n) for n = 0..1000</a>
%H A318256 András Bazsó and István Mező, <a href="http://dx.doi.org/10.1016/j.jnt.2015.01.019">On the coefficients of power sums of arithmetic progressions</a>, J. Number Th., 153 (2015), 117-123.
%H A318256 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 A318256 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 A318256 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 A318256 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.
%H A318256 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.
%F A318256 Let Q(n) = {p <= floor((n + 2)/(2 + n mod 2)) and p is prime and p does not divide n + 1 and the sum of the digits in base p of n+1 is at least p} then a(n) = Product_{p in Q(n)} p. (See the Kellner & Sondow links.)
%F A318256 a(n) = denominator(Bernoulli'(n+1, x)), where ' denotes d/dx. - _Peter Luschny_, Oct 15 2023
%e A318256 a(59) = 1 because there exist no number which satisfies the definition (and the product of an empty set is 1).
%e A318256 a(60) = 930930 because {2, 3, 5, 7, 11, 13, 31} are the only primes which satisfy the definition.
%e A318256 The denominator of the Bernoulli polynomial B_n(x) equals the squarefree kernel of n+1 if n is in {0, 1, 3, 5, 9, 11, 27, 29, 35, 59}. These might be the only numbers with this property.
%p A318256 a := n -> denom(bernoulli(n, x)) / mul(p, p in numtheory:-factorset(n+1)):
%p A318256 seq(a(n), n=0..61);
%t A318256 sfk[n_] := Times @@ FactorInteger[n][[All, 1]];
%t A318256 a[n_] := (BernoulliB[n, x] // Together // Denominator)/sfk[n+1];
%t A318256 Table[a[n], {n, 0, 61}] (* _Jean-François Alcover_, Feb 14 2019 *)
%o A318256 (Sage)
%o A318256 def A318256(n): return mul([p for p in (2..(n+2)//(2+n%2))
%o A318256                 if is_prime(p)
%o A318256                 and not p.divides(n+1)
%o A318256                 and sum((n+1).digits(base=p)) >= p])
%o A318256 print([A318256(n) for n in (0..61)])
%Y A318256 a(n) = A144845(n) / A007947(n+1).
%Y A318256 Cf. A324370 (same sequence with offset 1).
%Y A318256 Cf. A027642, A064538, A195441, A286515, A286516, A286517, A286762, A286763, A319084.
%K A318256 nonn
%O A318256 0,3
%A A318256 _Peter Luschny_, Sep 12 2018