A370814 Number of condensed integer factorizations of n into unordered factors > 1.

1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 4, 1, 2, 2, 4, 1, 4, 1, 4, 2, 2, 1, 6, 2, 2, 2, 4, 1, 5, 1, 5, 2, 2, 2, 8, 1, 2, 2, 6, 1, 5, 1, 4, 4, 2, 1, 10, 2, 4, 2, 4, 1, 6, 2, 6, 2, 2, 1, 11, 1, 2, 4, 7, 2, 5, 1, 4, 2, 5, 1, 14, 1, 2, 4, 4, 2, 5, 1, 10, 4, 2, 1, 11, 2

Offset: 1

Author

Gus Wiseman, Mar 04 2024

Keywords

• nonn

Comments

A multiset is condensed iff it is possible to choose a different divisor of each element.

Examples

			The a(36) = 7 factorizations: (2*2*9), (2*3*6), (2*18), (3*3*4), (3*12), (4*9), (6*6), (36).
		

Crossrefs

Partitions of this type are counted by A239312, ranks A368110.

Factors instead of divisors: A368414, complement A368413, unique A370645.

Partitions not of this type are counted by A370320, ranks A355740.

Subsets of this type: A370582 and A370636, complement A370583 and A370637.

The complement is counted by A370813, partitions A370593, ranks A355529.

For a unique choice we have A370815, partitions A370595, ranks A370810.

A000005 counts divisors.

A001055 counts factorizations, strict A045778.

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

Cf. A340596, A340653, A355739, A370592, A370803, A370804, A370805.

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[Select[Tuples[Divisors /@ #],UnsameQ@@#&]]>0&]],{n,100}]