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A322353 Number of factorizations of n into distinct semiprimes; a(1) = 1 by convention.

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