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A072931 Number of ways to write n as a sum of 2 semiprimes.

%I A072931 #38 Oct 19 2023 11:42:19
%S A072931 0,0,0,0,0,0,0,0,1,0,1,0,1,1,1,1,1,0,2,2,2,1,0,1,2,2,1,1,2,2,3,3,2,0,
%T A072931 1,3,3,2,1,3,3,2,3,4,4,2,1,4,5,3,3,1,3,3,2,5,3,2,2,5,6,6,1,3,5,3,4,4,
%U A072931 5,3,3,6,7,5,3,3,4,4,4,5,5,3,2,7,7,2,4,4,5,4,6,8,6,3,3,8,7,7,4,6,8,6,5,7,7,2
%N A072931 Number of ways to write n as a sum of 2 semiprimes.
%C A072931 Sequence is probably > 0 for n > 33.
%C A072931 The graph of this sequence is compelling evidence that 33 is the last term of sequence A072966. - _T. D. Noe_, Apr 10 2007
%H A072931 Alois P. Heinz, <a href="/A072931/b072931.txt">Table of n, a(n) for n = 0..20000</a> (first 10000 terms from T. D. Noe)
%F A072931 From _Reinhard Zumkeller_, Jan 21 2010: (Start)
%F A072931 a(A100592(n)) = n;
%F A072931 a(m) < n for m < A100592(n);
%F A072931 A171963(n) = a(A001358(n)). (End)
%F A072931 a(n) = Sum_{i=1..floor(n/2)} [Omega(i) == 2] * [Omega(n-i) == 2], where Omega = A001222 and [] is the Iverson Bracket. - _Wesley Ivan Hurt_, Apr 04 2018
%F A072931 a(n) = [x^n y^2] 1/Product_{j>=1} (1-y*x^A001358(j)). - _Alois P. Heinz_, May 21 2021
%t A072931 lim = 10000;
%t A072931 s = Select[Range[lim], PrimeOmega[#] == 2 &];
%t A072931 c = Tally[ Sort[ Map[ Total, Union[Subsets[s, {2}],
%t A072931       Table[{s[[i]], s[[i]]}, {i, 1, Length[s]}]]]]];
%t A072931 a = Table[0, lim];
%t A072931 i=1; While[c[[i]] [[1]]<=lim, a[[c[[i]] [[1]]]]=c[[i]] [[2]]; i++];
%t A072931 a (* _Robert Price_, Mar 30 2019 *)
%o A072931 (PARI) a(n)=sum(i=1, n, sum(j=1, i, if(abs(bigomega(i)-2) + abs(bigomega(j)-2) + abs(n-i-j),0,1)))
%o A072931 (PARI) a(n)=my(s); forprime(p=2,n\4, forprime(q=2,min(n\(2*p),p), if(bigomega(n-p*q)==2, s++))); s \\ _Charles R Greathouse IV_, Dec 07 2014
%Y A072931 Cf. A001222, A001358, A072966, A101048.
%Y A072931 Column k=2 of A344447.
%K A072931 easy,look,nonn
%O A072931 0,19
%A A072931 _Benoit Cloitre_, Aug 13 2002