login
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.
4

%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