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%I #14 Jan 05 2021 15:14:47
%S 1,1,1,1,1,2,1,1,1,2,1,2,1,2,2,1,1,2,1,2,2,2,1,2,1,2,1,2,1,4,1,1,2,2,
%T 2,3,1,2,2,2,1,4,1,2,2,2,1,2,1,2,2,2,1,2,2,2,2,2,1,5,1,2,2,1,2,4,1,2,
%U 2,4,1,3,1,2,2,2,2,4,1,2,1,2,1,5,2,2,2
%N Number of factorizations of n into primes or squarefree semiprimes.
%C A squarefree semiprime (A006881) is a product of any two distinct prime numbers.
%C Conjecture: also the number of semistandard Young tableaux whose entries are the prime indices of n (A323437).
%C Is this a duplicate of A323437? - _R. J. Mathar_, Jan 05 2021
%H Gus Wiseman, <a href="/A339741/a339741_1.txt">Counting and ranking factorizations, factorability, and vertex-degree partitions for groupings into pairs.</a>
%F a(A002110(n)) = A000085(n), and in general if n is a product of k distinct primes, a(n) = A000085(k).
%F a(n) = Sum_{d|n} A320656(n/d), so A320656 is the Moebius transform of this sequence.
%e The a(n) factorizations for n = 36, 60, 180, 360, 420, 840:
%e 6*6 6*10 5*6*6 6*6*10 2*6*35 6*10*14
%e 2*3*6 2*5*6 2*6*15 2*5*6*6 5*6*14 2*2*6*35
%e 2*2*3*3 2*2*15 3*6*10 2*2*6*15 6*7*10 2*5*6*14
%e 2*3*10 2*3*5*6 2*3*6*10 2*10*21 2*6*7*10
%e 2*2*3*5 2*2*3*15 2*2*3*5*6 2*14*15 2*2*10*21
%e 2*3*3*10 2*2*2*3*15 2*5*6*7 2*2*14*15
%e 2*2*3*3*5 2*2*3*3*10 3*10*14 2*2*5*6*7
%e 2*2*2*3*3*5 2*2*3*35 2*3*10*14
%e 2*2*5*21 2*2*2*3*35
%e 2*2*7*15 2*2*2*5*21
%e 2*3*5*14 2*2*2*7*15
%e 2*3*7*10 2*2*3*5*14
%e 2*2*3*5*7 2*2*3*7*10
%e 2*2*2*3*5*7
%t sqpe[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[sqpe[n/d],Min@@#>=d&]],{d,Select[Divisors[n],PrimeQ[#]||SquareFreeQ[#]&&PrimeOmega[#]==2&]}]];
%t Table[Length[sqpe[n]],{n,100}]
%Y See link for additional cross-references.
%Y Only allowing only primes gives A008966.
%Y Not allowing primes gives A320656.
%Y Unlabeled multiset partitions of this type are counted by A320663/A339888.
%Y Allowing squares of primes gives A320732.
%Y The strict version is A339742.
%Y A001055 counts factorizations.
%Y A001358 lists semiprimes, with squarefree case A006881.
%Y A002100 counts partitions into squarefree semiprimes.
%Y A338899/A270650/A270652 give the prime indices of squarefree semiprimes.
%Y Cf. A000070, A000961, A001221, A096373, A320893, A338914, A339740, A339741, A339841, A339846.
%K nonn
%O 1,6
%A _Gus Wiseman_, Dec 22 2020
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