A141056 1 followed by A027760, a variant of Bernoulli number denominators.

1, 2, 6, 2, 30, 2, 42, 2, 30, 2, 66, 2, 2730, 2, 6, 2, 510, 2, 798, 2, 330, 2, 138, 2, 2730, 2, 6, 2, 870, 2, 14322, 2, 510, 2, 6, 2, 1919190, 2, 6, 2, 13530, 2, 1806, 2, 690, 2, 282, 2, 46410, 2, 66, 2, 1590, 2, 798, 2, 870, 2, 354, 2, 56786730, 2, 6, 2, 510, 2, 64722, 2, 30, 2, 4686

Offset: 0

Author

Paul Curtz, Aug 01 2008

Keywords

• nonn

Comments

The denominators of the Bernoulli numbers for n>0. B_n sequence begins 1, -1/2, 1/6, 0/2, -1/30, 0/2, 1/42, 0/2, ... This is an alternative version of A027642 suggested by the theorem of Clausen. - Peter Luschny, Apr 29 2009

Let f(n,k) = gcd { multinomial(n; n1, ..., nk) | n1 + ... + nk = n }; then a(n) = f(N,N-n+1)/f(N,N-n) for N >> n. - Mamuka Jibladze, Mar 07 2017

Examples

			The rational values as given by the e.g.f. in the formula section start: 1, 1/2, 7/6, 3/2, 59/30, 5/2, 127/42, 7/2, 119/30, ... - _Peter Luschny_, Aug 18 2018
		

Links

• Antti Karttunen, Table of n, a(n) for n = 0..10080
• Thomas Clausen, Lehrsatz aus einer Abhandlung Über die Bernoullischen Zahlen, Astr. Nachr. 17 (22) (1840), 351-352.
• Wikipedia, Bernoulli number

Crossrefs

Cf. A027760, A027642, A176328.

Cf. A164555, A027642, A048993.

Programs

Maple

Clausen := proc(n) local S,i;
S := numtheory[divisors](n); S := map(i->i+1,S);
S := select(isprime,S); mul(i,i=S) end proc:
seq(Clausen(i),i=0..24);
# Peter Luschny, Apr 29 2009
A141056 := proc(n)
    if n = 0 then 1 else A027760(n) end if;
end proc: # R. J. Mathar, Oct 28 2013

Mathematica

a[n_] := Sum[ Boole[ PrimeQ[d+1]] / (d+1), {d, Divisors[n]}] // Denominator; Table[a[n], {n, 0, 70}] (* Jean-François Alcover, Aug 09 2012 *)

PARI

A141056(n) =
{
    p = 1;
    if (n > 0,
        fordiv(n, d,
            r = d + 1;
            if (isprime(r), p = p*r)
        )
    );
    return(p)
}
for(n=0,70,print1(A141056(n), ", ")); /* Peter Luschny, May 07 2012 */

Formula

a(n) are the denominators of the polynomials generated by cosh(x*z)*z/(1-exp(-z)) evaluated x=1. See A176328 for the numerators. - Peter Luschny, Aug 18 2018

a(n) = denominator(Sum_{j=0..n} (-1)^(n-j)*j!*Stirling2(n,j)*B(j)), where B are the Bernoulli numbers A164555/A027642. - Fabián Pereyra, Jan 06 2022

Extensions

Extended by R. J. Mathar, Nov 22 2009