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A002100 a(n) = number of partitions of n into semiprimes (more precisely, number of ways of writing n as a sum of products of 2 distinct primes).

%I A002100 M0205 N0076 #26 Nov 26 2020 12:19:28
%S A002100 0,0,0,0,0,1,0,0,0,1,0,1,0,1,1,1,0,1,0,2,2,2,0,2,1,3,2,3,1,4,2,4,3,5,
%T A002100 4,7,3,6,5,8,6,10,6,10,9,12,9,15,11,16,14,18,14,22,19,25,22,27,23,33,
%U A002100 29,36,33,40,38,49,43,53,51,61,57,71,64,77,76,89,86,102,96,113,111,128,125
%N A002100 a(n) = number of partitions of n into semiprimes (more precisely, number of ways of writing n as a sum of products of 2 distinct primes).
%D A002100 L. M. Chawla and S. A. Shad, On a restricted partition function t(n) and its table, J. Natural Sciences and Mathematics, 9 (1969), 217-221. Math. Rev. 41 #6761.
%D A002100 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A002100 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A002100 T. D. Noe, <a href="/A002100/b002100.txt">Table of n, a(n) for n = 1..1000</a>
%e A002100 a(20) = 2: 20 = 2*3 + 2*7 = 2*5 + 2*5.
%t A002100 a[n_] := SeriesCoefficient[1/Product[If[SquareFreeQ[k] && PrimeNu[k] == 2, 1 - z^k, 1], {k, 1, n}], {z, 0, n}];
%t A002100 Array[a, 100] (* _Jean-François Alcover_, Nov 26 2020, after PARI *)
%o A002100 (PARI) a(n)=polcoeff(1/prod(k=1,n,if(issquarefree(k)*if(omega(k)-2,0,1),1-z^k,1))+O(z^(n+1)),n)
%o A002100 (Haskell)
%o A002100 a002100 = p a006881_list where
%o A002100    p _          0 = 1
%o A002100    p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m
%o A002100 -- _Reinhard Zumkeller_, Mar 21 2014
%Y A002100 Cf. A006881, A073576, A101048.
%K A002100 nonn
%O A002100 1,20
%A A002100 _N. J. A. Sloane_
%E A002100 More terms from _Benoit Cloitre_, Jun 01 2003