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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. 6
 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, 14, 1, 30, 1, 462, 231, 1190, 105, 6, 1, 51870, 1365, 70, 21, 2310, 55, 2310, 105, 322, 105, 210, 35, 6630, 663, 286, 33, 330, 55, 798, 57, 290, 15, 30, 1, 930930, 15015 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 LINKS Peter Luschny, Table of n, a(n) for n = 0..1000 András Bazsó and István Mező, On the coefficients of power sums of arithmetic progressions, J. Number Th., 153 (2015), 117-123. Bernd C. Kellner, On a product of certain primes, J. Number Theory, 179 (2017), 126-141; arXiv:1705.04303 [math.NT], 2017. Bernd C. Kellner and Jonathan Sondow, Power-Sum Denominators, Amer. Math. Monthly, 124 (2017), 695-709; arXiv:1705.03857 [math.NT], 2017. Bernd C. Kellner and Jonathan Sondow, The denominators of power sums of arithmetic progressions, Integers 18 (2018), #A95, 17 pp.; arXiv:1705.05331 [math.NT], 2017. Bernd C. Kellner and Jonathan Sondow, On Carmichael and polygonal numbers, Bernoulli polynomials, and sums of base-p digits, Integers 21 (2021), #A52, 21 pp.; arXiv:1902.10672 [math.NT], 2019. Bernd C. Kellner, On the finiteness of Bernoulli polynomials whose derivative has only integral coefficients, 9 pp.; arXiv:2310.01325 [math.NT], 2023. FORMULA 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.) a(n) = denominator(Bernoulli'(n+1, x)), where ' denotes d/dx. - Peter Luschny, Oct 15 2023 EXAMPLE a(59) = 1 because there exist no number which satisfies the definition (and the product of an empty set is 1). a(60) = 930930 because {2, 3, 5, 7, 11, 13, 31} are the only primes which satisfy the definition. 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. MAPLE a := n -> denom(bernoulli(n, x)) / mul(p, p in numtheory:-factorset(n+1)): seq(a(n), n=0..61); MATHEMATICA sfk[n_] := Times @@ FactorInteger[n][[All, 1]]; a[n_] := (BernoulliB[n, x] // Together // Denominator)/sfk[n+1]; Table[a[n], {n, 0, 61}] (* Jean-François Alcover, Feb 14 2019 *) PROG (Sage) def A318256(n): return mul([p for p in (2..(n+2)//(2+n%2)) if is_prime(p) and not p.divides(n+1) and sum((n+1).digits(base=p)) >= p]) print([A318256(n) for n in (0..61)]) CROSSREFS a(n) = A144845(n) / A007947(n+1). Cf. A324370 (same sequence with offset 1). Cf. A027642, A064538, A195441, A286515, A286516, A286517, A286762, A286763, A319084. Sequence in context: A366570 A286515 A166120 * A324370 A324193 A364829 Adjacent sequences: A318253 A318254 A318255 * A318257 A318258 A318259 KEYWORD nonn AUTHOR Peter Luschny, Sep 12 2018 STATUS approved

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Last modified May 18 12:18 EDT 2024. Contains 372630 sequences. (Running on oeis4.)