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A339887 Number of factorizations of n into primes or squarefree semiprimes.

%I A339887 #14 Jan 05 2021 15:14:47
%S A339887 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 A339887 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 A339887 2,4,1,3,1,2,2,2,2,4,1,2,1,2,1,5,2,2,2
%N A339887 Number of factorizations of n into primes or squarefree semiprimes.
%C A339887 A squarefree semiprime (A006881) is a product of any two distinct prime numbers.
%C A339887 Conjecture: also the number of semistandard Young tableaux whose entries are the prime indices of n (A323437).
%C A339887 Is this a duplicate of A323437? - _R. J. Mathar_, Jan 05 2021
%H A339887 Gus Wiseman, <a href="/A339741/a339741_1.txt">Counting and ranking factorizations, factorability, and vertex-degree partitions for groupings into pairs.</a>
%F A339887 a(A002110(n)) = A000085(n), and in general if n is a product of k distinct primes, a(n) = A000085(k).
%F A339887 a(n) = Sum_{d|n} A320656(n/d), so A320656 is the Moebius transform of this sequence.
%e A339887 The a(n) factorizations for n = 36, 60, 180, 360, 420, 840:
%e A339887   6*6       6*10      5*6*6       6*6*10        2*6*35      6*10*14
%e A339887   2*3*6     2*5*6     2*6*15      2*5*6*6       5*6*14      2*2*6*35
%e A339887   2*2*3*3   2*2*15    3*6*10      2*2*6*15      6*7*10      2*5*6*14
%e A339887             2*3*10    2*3*5*6     2*3*6*10      2*10*21     2*6*7*10
%e A339887             2*2*3*5   2*2*3*15    2*2*3*5*6     2*14*15     2*2*10*21
%e A339887                       2*3*3*10    2*2*2*3*15    2*5*6*7     2*2*14*15
%e A339887                       2*2*3*3*5   2*2*3*3*10    3*10*14     2*2*5*6*7
%e A339887                                   2*2*2*3*3*5   2*2*3*35    2*3*10*14
%e A339887                                                 2*2*5*21    2*2*2*3*35
%e A339887                                                 2*2*7*15    2*2*2*5*21
%e A339887                                                 2*3*5*14    2*2*2*7*15
%e A339887                                                 2*3*7*10    2*2*3*5*14
%e A339887                                                 2*2*3*5*7   2*2*3*7*10
%e A339887                                                             2*2*2*3*5*7
%t A339887 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 A339887 Table[Length[sqpe[n]],{n,100}]
%Y A339887 See link for additional cross-references.
%Y A339887 Only allowing only primes gives A008966.
%Y A339887 Not allowing primes gives A320656.
%Y A339887 Unlabeled multiset partitions of this type are counted by A320663/A339888.
%Y A339887 Allowing squares of primes gives A320732.
%Y A339887 The strict version is A339742.
%Y A339887 A001055 counts factorizations.
%Y A339887 A001358 lists semiprimes, with squarefree case A006881.
%Y A339887 A002100 counts partitions into squarefree semiprimes.
%Y A339887 A338899/A270650/A270652 give the prime indices of squarefree semiprimes.
%Y A339887 Cf. A000070, A000961, A001221, A096373, A320893, A338914, A339740, A339741, A339841, A339846.
%K A339887 nonn
%O A339887 1,6
%A A339887 _Gus Wiseman_, Dec 22 2020