A370815 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.

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, 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, 0, 0, 0, 2, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0

Offset: 1

Author

Gus Wiseman, Mar 06 2024

Keywords

• nonn

Examples

			The a(432) = 3 factorizations: (2*2*3*4*9), (2*3*3*4*6), (2*6*6*6).
		

Crossrefs

For partitions and prime factors we have A370594, ranks A370647.

Partitions of this type are counted by A370595, ranks A370810.

For prime factors we have A370645, subsets A370584.

A000005 counts divisors.

A001055 counts factorizations, strict A045778.

A239312 counts condensed partitions, ranks A355740, complement A370320.

A355731 counts choices of a divisor of each prime index, firsts A355732.

A368414 counts factor-choosable factorizations, complement A368413.

A370814 counts divisor-choosable factorizations, complement A370813.

Cf. A340596, A340653, A355529, A355739, A368110, A370592, A370638, A370803.

Programs

Mathematica

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&, Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[facs[n],Length[Union[Sort /@ Select[Tuples[Divisors/@#],UnsameQ@@#&]]]==1&]],{n,100}]