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A141056
1 followed by A027760, a variant of Bernoulli number denominators.
32
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
0,2
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
LINKS
Thomas Clausen, Lehrsatz aus einer Abhandlung Über die Bernoullischen Zahlen, Astr. Nachr. 17 (22) (1840), 351-352.
Wikipedia, Bernoulli number
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
EXAMPLE
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
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 *)
PROG
(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 */
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul Curtz, Aug 01 2008
EXTENSIONS
Extended by R. J. Mathar, Nov 22 2009
STATUS
approved