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A101048 Number of partitions of n into semiprimes (a(0) = 1 by convention).

%I A101048 #33 Jan 20 2023 16:00:28
%S A101048 1,0,0,0,1,0,1,0,1,1,2,0,2,1,3,2,3,1,5,3,5,4,7,4,9,7,10,8,13,10,17,13,
%T A101048 18,17,25,21,29,25,34,34,43,37,51,49,61,59,73,69,89,87,103,103,124,
%U A101048 122,148,149,172,176,206,208,244,248,281,293,337,344,391,405,456,479,537,553
%N A101048 Number of partitions of n into semiprimes (a(0) = 1 by convention).
%C A101048 Semiprime analog of A000607. a(n) <= A002095(n). - _Jonathan Vos Post_, Oct 01 2007
%C A101048 Das, Robles, Zaharescu, & Zeindler give an asymptotic formula, see Links. - _Charles R Greathouse IV_, Jan 20 2023
%H A101048 Alois P. Heinz, <a href="/A101048/b101048.txt">Table of n, a(n) for n = 0..10000</a> (first 1001 terms from T. D. Noe)
%H A101048 Madhuparna Das, Nicolas Robles, Alexandru Zaharescu, and Dirk Zeindler, <a href="https://arxiv.org/abs/2212.12489">Partitions into semiprimes</a>, arXiv preprint (2022). arXiv:2212.12489 [math.NT]
%F A101048 G.f.: 1/product(product(1-x^(p(i)p(j)), i = 1..j),j = 1..infinity), p(k) is the k-th prime. - _Emeric Deutsch_, Apr 04 2006
%e A101048 a(12) = #{6 + 6, 4 + 4 + 4} = #{2 * (2*3), 3 * (2*2)} = 2.
%p A101048 g:=1/product(product(1-x^(ithprime(i)*ithprime(j)),i=1..j),j=1..30): gser:=series(g,x=0,75): seq(coeff(gser,x,n),n=1..71); # _Emeric Deutsch_, Apr 04 2006
%p A101048 # second Maple program:
%p A101048 h:= proc(n) option remember; `if`(n=0, 0,
%p A101048      `if`(numtheory[bigomega](n)=2, n, h(n-1)))
%p A101048     end:
%p A101048 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A101048      `if`(i>n, 0, b(n-i, h(min(n-i, i))))+b(n, h(i-1))))
%p A101048     end:
%p A101048 a:= n-> b(n, h(n)):
%p A101048 seq(a(n), n=0..100);  # _Alois P. Heinz_, May 19 2021
%t A101048 terms = 100; CoefficientList[1/Product[1 - x^(Prime[i] Prime[j]), {i, 1, PrimePi[Ceiling[terms/2]]}, {j, 1, i}] + O[x]^terms, x] (* _Jean-François Alcover_, Aug 01 2018 *)
%o A101048 (Haskell)
%o A101048 a101048 = p a001358_list where
%o A101048    p _          0 = 1
%o A101048    p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m
%o A101048 -- _Reinhard Zumkeller_, Mar 21 2014
%o A101048 (PARI) issemi(n)=if(n<4, return(0)); forprime(p=2,97, if(n%p==0, return(isprime(n/p)))); bigomega(n)==2
%o A101048 allsemi(v)=for(i=1,#v, if(!issemi(v[i]), return(0))); 1
%o A101048 a(n)=my(s); if(n<4, return(n==0)); forpart(k=n, if(allsemi(k), s++),[4,n]); s \\ _Charles R Greathouse IV_, Jan 20 2023
%Y A101048 Cf. A000041, A000607, A101049, A001358, A064911, A002095.
%Y A101048 Cf. A112020, A112021.
%Y A101048 Cf. A002100.
%Y A101048 Row sums of A344447.
%K A101048 nonn
%O A101048 0,11
%A A101048 _Reinhard Zumkeller_, Nov 28 2004
%E A101048 a(0) set to 1 by _N. J. A. Sloane_, Nov 23 2007