cp's OEIS Frontend

A116357 Number of partitions of n into products of two successive primes (A006094).

%I A116357 #18 Sep 11 2024 02:24:25
%S A116357 0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,1,0,0,1,0,0,1,0,0,1,0,0,2,0,0,1,0,
%T A116357 1,2,0,0,1,0,1,2,0,0,2,0,1,2,0,1,2,0,1,2,0,1,2,0,1,3,0,1,2,0,2,3,0,1,
%U A116357 2,1,2,3,0,1,3,1,3,3,0,2,3,1,3,3,1,2,3,1,3,4,1,3,3,1,4,4,1,3,3,2,4,4,1,3,5
%N A116357 Number of partitions of n into products of two successive primes (A006094).
%C A116357 a(A116358(n)) = 0; a(A116359(n)) > 0;
%C A116357 a(n) < A101048(n).
%H A116357 Robert Israel, <a href="/A116357/b116357.txt">Table of n, a(n) for n = 1..10000</a>
%F A116357 G.f.: Product_{k >= 1} 1/(1 - x^(prime(k)*prime(k+1))). - _Robert Israel_, Dec 09 2016
%e A116357 a(41) = #{2*3 + 5*7} = 1;
%e A116357 a(42) = #{2*3+2*3+2*3+2*3+2*3+2*3+2*3, 2*3+2*3+3*5+3*5} = 2.
%p A116357 N:= 200: # to get a(1) to a(N)
%p A116357 Primes:= select(isprime,[2,seq(i,i=3..1+floor(sqrt(N)),2)]):
%p A116357 G:= mul(1/(1 - x^(Primes[i]*Primes[i+1])), i=1..nops(Primes)-1):
%p A116357 S:= series(G,x,N+1):
%p A116357 seq(coeff(S,x,j),j=1..N); # _Robert Israel_, Dec 09 2016
%t A116357 m = 105; kmax = PrimePi[Sqrt[m]]; Product[1/(1-x^(Prime[k]*Prime[k+1])), {k, 1, kmax}] + O[x]^(m+1) // CoefficientList[#, x]& // Rest (* _Jean-François Alcover_, Mar 09 2019, after _Robert Israel_ *)
%Y A116357 Cf. A006094, A101048, A116358, A116359, A116360.
%K A116357 nonn
%O A116357 1,30
%A A116357 _Reinhard Zumkeller_, Feb 12 2006