A370813 Number of non-condensed integer factorizations of n into unordered factors > 1.

0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 0, 0

Offset

1,32

Comments

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

Links

Table of n, a(n) for n=1..87.

Example

The a(96) = 4 factorizations: (2*2*2*2*2*3), (2*2*2*2*6), (2*2*2*3*4), (2*2*2*12).

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}]

Crossrefs

Partitions not of this type are counted by A239312, ranks A368110.

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

Partitions of this type are counted by A370320, ranks A355740.

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

The complement is counted by A370814, partitions A370592, ranks A368100.

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, A355529, A355739, A355741, A370804, A370807.

Sequence in context: A287619 A133698 A219488 * A129251 A276077 A276935

Adjacent sequences: A370810 A370811 A370812 * A370814 A370815 A370816

Keyword

nonn

Author

Gus Wiseman, Mar 04 2024

Status

approved