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 A141459 a(n) = Product_{p-1 divides n} p, where p is an odd prime. 14
 1, 1, 3, 1, 15, 1, 21, 1, 15, 1, 33, 1, 1365, 1, 3, 1, 255, 1, 399, 1, 165, 1, 69, 1, 1365, 1, 3, 1, 435, 1, 7161, 1, 255, 1, 3, 1, 959595, 1, 3, 1, 6765, 1, 903, 1, 345, 1, 141, 1, 23205, 1, 33, 1, 795, 1, 399, 1, 435, 1, 177, 1, 28393365, 1, 3, 1, 255, 1, 32361, 1, 15, 1, 2343, 1, 70050435 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Previous name was: A027760(n)/2 for n>=1, a(0) = 1. Conjecture: a(n) = denominator of integral_{0..1}(log(1-1/x)^n) dx. - Jean-François Alcover, Feb 01 2013 Define the generalized Bernoulli function as B(s,z) = -s*z^s*HurwitzZeta(1-s,1/z) for Re(1/z) > 0 and B(0,z) = 1 for all z; further the generalized Bernoulli polynomials as Bp(m,n,z) = Sum_{j=0..n} B(j,m)*C(n,j)*(z-1)^(n-j) then the a(n) are denominators of Bp(2,n,1), i. e. of the generalized Bernoulli numbers in the case m=2. The numerators of these numbers are A157779(n). - Peter Luschny, May 17 2015 From Peter Luschny, Nov 22 2015: (Start) a(n) are the denominators of the centralized Bernoulli polynomials 2^n*Bernoulli(n, x/2+1/2) evaluated at x=1. The numerators are A239275(n). a(n) is the odd part of A141056(n). a(n) is squarefree, by the von Staudt-Clausen theorem. (End) LINKS Robert Israel, Table of n, a(n) for n = 0..10000 Peter Luschny, Generalized Bernoulli Numbers and Polynomials FORMULA a(2*n+1) = 1. a(2*n)= A001897(n). a(n) = denominator(0^n + Sum_{j=1..n} zeta(1-j)*(2^j-2)*j*C(n,j)). - Peter Luschny, May 17 2015 Let P(x)= Sum_{n>=0} x^(2*n+1)/(2*n+1)! then a(n) = denominator( n! [x^n] x/P(x) ). - Peter Luschny, Jul 05 2016 a(n) = A157818(n)/4^n. See a comment under A157817, also for other Bernoulli numbers B[4,1] and B[4,3] with this denominator. - Wolfdieter Lang, Apr 28 2017 EXAMPLE The denominators of 1, 0, -1/3, 0, 7/15, 0, -31/21, 0, 127/15, 0, -2555/33, 0, 1414477/1365, ... MAPLE Bfun := (s, z) -> `if`(s=0, 1, -s*z^s*Zeta(0, 1-s, 1/z): # generalized Bernoulli function Bpoly := (m, n, z) -> add(Bfun(j, m)*binomial(n, j)*(z-1)^(n-j), j=0..n): # generalized Bernoulli polynomials seq(Bpoly(2, n, 1), n=0..50): denom([%]); # which simplifies to: a := n -> 0^n+add(Zeta(1-j)*(2^j-2)*j*binomial(n, j), j=1..n): seq(denom(a(n)), n=0..50); # Peter Luschny, May 17 2015 # Alternatively: with(numtheory): ClausenOdd := proc(n) local S, m; S := map(i -> i + 1, divisors(n)); S := select(isprime, S) minus {2}; mul(m, m = S) end: seq(ClausenOdd(n), n=0..72); # Peter Luschny, Nov 22 2015 # Alternatively: N:= 1000: # to get a(0) to a(N) V:= Array(0..N, 1): for p in select(isprime, [seq(i, i=3..N+1, 2)]) do   R:=[seq(j, j=p-1..N, p-1)]:   V[R]:= V[R] * p; od: convert(V, list); # Robert Israel, Nov 22 2015 MATHEMATICA a[n_] := If[OddQ[n], 1, Denominator[-2*(2^(n - 1) - 1)*BernoulliB[n]]]; Table[a[n], {n, 0, 72}] (* Jean-François Alcover, Jan 30 2013 *) Table[Times @@ Select[Divisors@ n + 1, PrimeQ@ # && OddQ@ # &] + Boole[n == 0], {n, 0, 72}] (* Michael De Vlieger, Apr 30 2017 *) PROG (PARI) A141056(n) = {     p = 1;     if (n > 0,         fordiv(n, d,             r = d + 1;             if (isprime(r) & r>2, p = p*r)         )     );     return(p) } for(n=0, 72, print1(A141056(n), ", ")); \\ Peter Luschny, Nov 22 2015 (Sage) def A141459_list(size):     f = x / sum(x^(n*2+1)/factorial(n*2+1) for n in (0..2*size))     t = taylor(f, x, 0, size)     return [(factorial(n)*s).denominator() for n, s in enumerate (t.list())] print A141459_list(72) # Peter Luschny, Jul 05 2016 CROSSREFS Cf. A027760, A141056, A141459, A157779, A157818, A160014, A226157, A239275. Sequence in context: A286768 A286114 A214073 * A318142 A176727 A080924 Adjacent sequences:  A141456 A141457 A141458 * A141460 A141461 A141462 KEYWORD nonn AUTHOR Paul Curtz, Aug 08 2008 EXTENSIONS 1 prepended and offset set to 0 by Peter Luschny, May 17 2015 New name from Peter Luschny, Nov 22 2015 STATUS approved

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Last modified June 24 09:37 EDT 2019. Contains 324323 sequences. (Running on oeis4.)