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%I #20 Jan 05 2021 21:34:43
%S 1,0,0,1,0,1,0,0,1,1,0,0,0,1,1,0,0,0,0,0,1,1,0,1,1,1,0,0,0,0,0,0,1,1,
%T 1,1,0,1,1,1,0,0,0,0,0,1,0,0,1,0,1,0,0,1,1,1,1,1,0,2,0,1,0,0,1,0,0,0,
%U 1,0,0,0,0,1,0,0,1,0,0,0,0,1,0,2,1,1,1,1,0,2,1,0,1,1,1,0,0,0,0,1,0,0,0,1,0
%N Number of factorizations of n into distinct semiprimes; a(1) = 1 by convention.
%C A semiprime (A001358) is a product of any two prime numbers. In the even case, these factorizations have A001222(n)/2 factors. - _Gus Wiseman_, Dec 31 2020
%C Records 1, 2, 3, 4, 5, 9, 13, 15, 17, ... occur at 1, 60, 210, 840, 1260, 4620, 27720, 30030, 69300, ...
%H Antti Karttunen, <a href="/A322353/b322353.txt">Table of n, a(n) for n = 1..100000</a>
%H Gus Wiseman, <a href="/A339741/a339741_1.txt">Counting and ranking factorizations, factorability, and vertex-degree partitions for groupings into pairs.</a>
%H <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>
%F a(n) = Sum_{d|n} (-1)^A001222(d) * A339839(n/d). - _Gus Wiseman_, Dec 31 2020
%e a(4) = 1, as there is just one way to factor 4 into distinct semiprimes, namely as {4}.
%e From _Gus Wiseman_, Dec 31 2020: (Start)
%e The a(n) factorizations for n = 60, 210, 840, 1260, 4620, 12600, 18480:
%e 4*15 6*35 4*6*35 4*9*35 4*15*77 4*6*15*35 4*6*10*77
%e 6*10 10*21 4*10*21 4*15*21 4*21*55 4*6*21*25 4*6*14*55
%e 14*15 4*14*15 6*10*21 4*33*35 4*9*10*35 4*6*22*35
%e 6*10*14 6*14*15 6*10*77 4*9*14*25 4*10*14*33
%e 9*10*14 6*14*55 4*10*15*21 4*10*21*22
%e 6*22*35 6*10*14*15 4*14*15*22
%e 10*14*33 6*10*14*22
%e 10*21*22
%e 14*15*22
%e (End)
%t Table[Count[Subsets[Select[Divisors[n], PrimeOmega[#] == 2 &]], _?(Times @@ # == n &)], {n, 105}] (* _Michael De Vlieger_, Dec 11 2020 *)
%o (PARI) A322353(n, m=n) = if(1==n, 1, my(s=0); fordiv(n, d, if((2==bigomega(d)&&(d<=m)), s += A322353(n/d, d-1))); (s)); \\ _Antti Karttunen_, Dec 10 2020
%Y Unlabeled multiset partitions of this type are counted by A007717.
%Y The version for partitions is A112020, or A101048 without distinctness.
%Y The non-strict version is A320655.
%Y Positions of zeros include A320892.
%Y Positions of nonzero terms are A320912.
%Y The case of squarefree factors is A339661, or A320656 without distinctness.
%Y Allowing prime factors gives A339839, or A320732 without distinctness.
%Y A322661 counts loop-graphs, ranked by A320461.
%Y A001055 counts factorizations, with strict case A045778.
%Y A001358 lists semiprimes, with squarefree case A006881.
%Y A027187 counts partitions of even length, ranked by A028260.
%Y A037143 lists primes and semiprimes.
%Y A338898/A338912/A338913 give the prime indices of semiprimes.
%Y A339846 counts even-length factorizations, with ordered version A174725.
%Y Cf. A001221, A006125, A006129, A028260, A320893, A338915, A339841.
%K nonn
%O 1,60
%A _Antti Karttunen_, Dec 06 2018
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