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%I #48 Feb 18 2022 23:08:49
%S 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,
%T 2,6,2,870,2,14322,2,510,2,6,2,1919190,2,6,2,13530,2,1806,2,690,2,282,
%U 2,46410,2,66,2,1590,2,798,2,870,2,354,2,56786730,2,6,2,510,2,64722,2,30,2,4686
%N 1 followed by A027760, a variant of Bernoulli number denominators.
%C 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
%C 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
%H Antti Karttunen, <a href="/A141056/b141056.txt">Table of n, a(n) for n = 0..10080</a>
%H Thomas Clausen, <a href="http://adsabs.harvard.edu/abs/1840AN.....17R.351">Lehrsatz aus einer Abhandlung Über die Bernoullischen Zahlen</a>, Astr. Nachr. 17 (22) (1840), 351-352.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Bernoulli_number">Bernoulli number</a>
%F 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
%F 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
%e 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
%p Clausen := proc(n) local S,i;
%p S := numtheory[divisors](n); S := map(i->i+1,S);
%p S := select(isprime,S); mul(i,i=S) end proc:
%p seq(Clausen(i),i=0..24);
%p # _Peter Luschny_, Apr 29 2009
%p A141056 := proc(n)
%p if n = 0 then 1 else A027760(n) end if;
%p end proc: # _R. J. Mathar_, Oct 28 2013
%t 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 *)
%o (PARI)
%o A141056(n) =
%o {
%o p = 1;
%o if (n > 0,
%o fordiv(n, d,
%o r = d + 1;
%o if (isprime(r), p = p*r)
%o )
%o );
%o return(p)
%o }
%o for(n=0,70,print1(A141056(n), ", ")); /* _Peter Luschny_, May 07 2012 */
%Y Cf. A027760, A027642, A176328.
%Y Cf. A164555, A027642, A048993.
%K nonn
%O 0,2
%A _Paul Curtz_, Aug 01 2008
%E Extended by _R. J. Mathar_, Nov 22 2009
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