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%I #8 Mar 03 2024 21:01:39
%S 1,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,1,2,1,2,1,1,1,3,1,1,1,2,1,1,1,1,1,1,
%T 1,4,1,1,1,3,1,1,1,2,2,1,1,4,1,2,1,2,1,3,1,3,1,1,1,3,1,1,2,1,1,1,1,2,
%U 1,1,1,5,1,1,2,2,1,1,1,4,1,1,1,3,1,1,1
%N Number of integer factorizations of n into unordered factors > 1 such that only one set can be obtained by choosing a different prime factor of each factor.
%C All of these factorizations are co-balanced (A340596).
%e The factorization f = (3*6*10) has prime factor choices (3,2,2), (3,3,2), (3,2,5), and (3,3,5), of which only (3,2,5) has all different parts, so f is counted under a(180).
%e The a(n) factorizations for n = 2, 12, 24, 36, 72, 120, 144, 180, 288:
%e (2) (2*6) (3*8) (4*9) (8*9) (3*5*8) (2*72) (4*5*9) (3*96)
%e (3*4) (4*6) (6*6) (2*36) (4*5*6) (3*48) (5*6*6) (4*72)
%e (2*12) (2*18) (3*24) (2*3*20) (4*36) (2*3*30) (6*48)
%e (3*12) (4*18) (2*5*12) (6*24) (2*5*18) (8*36)
%e (6*12) (2*6*10) (8*18) (2*6*15) (9*32)
%e (3*4*10) (9*16) (2*9*10) (12*24)
%e (12*12) (3*4*15) (16*18)
%e (3*5*12) (2*144)
%e (3*6*10)
%t facs[n_]:=If[n<=1,{{}},Join @@ Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]], {d,Rest[Divisors[n]]}]];
%t Table[Length[Select[facs[n], Length[Union[Sort/@Select[Tuples[First /@ FactorInteger[#]&/@#], UnsameQ@@#&]]]==1&]],{n,100}]
%Y Multisets of this type are ranked by A368101, see also A368100, A355529.
%Y For nonexistence we have A368413, complement A368414.
%Y Subsets of this type are counted by A370584, see also A370582, A370583.
%Y Maximal sets of this type are counted by A370585.
%Y The version for partitions is A370594, see also A370592, A370593.
%Y Subsets of this type are counted by A370638, see also A370636, A370637.
%Y For unlabeled multiset partitions we have A370646, also A368098, A368097.
%Y A001055 counts factorizations, strict A045778.
%Y A006530 gives greatest prime factor, least A020639.
%Y A007716 counts non-isomorphic multiset partitions, connected A007718.
%Y A027746 lists prime factors, A112798 indices, length A001222.
%Y A058891 counts set-systems, unlabeled A000612, connected A323818.
%Y A283877 counts non-isomorphic set-systems, connected A300913.
%Y A355741 counts ways to choose a prime factor of each prime index.
%Y For set-systems see A367902-A367908.
%Y Cf. A000040, A000720, A003963, A340596, A340653, A355744, A355745, A368110.
%K nonn
%O 1,12
%A _Gus Wiseman_, Mar 01 2024
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