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%I #6 Mar 06 2024 14:48:33
%S 1,0,0,1,0,0,0,0,1,0,0,1,0,0,0,1,0,1,0,1,0,0,0,0,1,0,0,1,0,0,0,1,0,0,
%T 0,0,0,0,0,0,0,0,0,1,1,0,0,1,1,1,0,1,0,0,0,0,0,0,0,1,0,0,1,1,0,0,0,1,
%U 0,0,0,2,0,0,1,1,0,0,0,1,1,0,0,1,0,0,0
%N Number of integer factorizations of n into unordered factors > 1, such that only one set can be obtained by choosing a different divisor of each factor.
%e The a(432) = 3 factorizations: (2*2*3*4*9), (2*3*3*4*6), (2*6*6*6).
%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[Divisors/@#],UnsameQ@@#&]]]==1&]],{n,100}]
%Y For partitions and prime factors we have A370594, ranks A370647.
%Y Partitions of this type are counted by A370595, ranks A370810.
%Y For prime factors we have A370645, subsets A370584.
%Y A000005 counts divisors.
%Y A001055 counts factorizations, strict A045778.
%Y A239312 counts condensed partitions, ranks A355740, complement A370320.
%Y A355731 counts choices of a divisor of each prime index, firsts A355732.
%Y A368414 counts factor-choosable factorizations, complement A368413.
%Y A370814 counts divisor-choosable factorizations, complement A370813.
%Y Cf. A340596, A340653, A355529, A355739, A368110, A370592, A370638, A370803.
%K nonn
%O 1,72
%A _Gus Wiseman_, Mar 06 2024
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