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A141459 a(n) = Product_{p-1 divides n} p, where p is an odd prime. 8
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

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 *)

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, A160014, A226157, A239275.

Sequence in context: A101820 A055301 A214073 * A176727 A080924 A232179

Adjacent sequences:  A141456 A141457 A141458 * A141460 A141461 A141462

KEYWORD

nonn

AUTHOR

Paul Curtz, Aug 08 2008

EXTENSIONS

Prepended 1 and set offset to 0 - Peter Luschny, May 17 2015

New name from Peter Luschny, Nov 22 2015

STATUS

approved

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Last modified December 4 23:10 EST 2016. Contains 278755 sequences.