A383310 Number of ways to choose a strict multiset partition of a factorization of n into factors > 1.

1, 1, 1, 2, 1, 3, 1, 5, 2, 3, 1, 8, 1, 3, 3, 9, 1, 8, 1, 8, 3, 3, 1, 20, 2, 3, 5, 8, 1, 12, 1, 19, 3, 3, 3, 24, 1, 3, 3, 20, 1, 12, 1, 8, 8, 3, 1, 46, 2, 8, 3, 8, 1, 20, 3, 20, 3, 3, 1, 38, 1, 3, 8, 37, 3, 12, 1, 8, 3, 12, 1, 67, 1, 3, 8, 8, 3, 12, 1, 46, 9, 3

Offset

1,4

Links

Table of n, a(n) for n=1..82.

Example

The a(36) = 24 choices:
  {{2,2,3,3}}  {{2},{2,3,3}}  {{2},{3},{2,3}}
  {{2,2,9}}    {{3},{2,2,3}}  {{2},{3},{6}}
  {{2,3,6}}    {{2,2},{3,3}}
  {{2,18}}     {{2},{2,9}}
  {{3,3,4}}    {{9},{2,2}}
  {{3,12}}     {{2},{3,6}}
  {{4,9}}      {{3},{2,6}}
  {{6,6}}      {{6},{2,3}}
  {{36}}       {{2},{18}}
               {{3},{3,4}}
               {{4},{3,3}}
               {{3},{12}}
               {{4},{9}}

Mathematica

facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];
sps[{}]:={{}}; sps[set:{i_, ___}]:=Join@@Function[s, Prepend[#, s]&/@sps[Complement[set, s]]]/@Cases[Subsets[set], {i, ___}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
Table[Sum[Length[Select[mps[y], UnsameQ@@#&]], {y, facs[n]}], {n, 30}]

Crossrefs

The case of a unique choice (positions of 1) is A008578.

This is the strict case of A050336.

For distinct strict blocks we have A050345.

For integer partitions we have A261049, strict case of A001970.

For strict blocks that are not necessarily distinct we have A296119.

Twice-partitions of this type are counted by A296122.

For normal multisets we have A317776, strict case of A255906.

A001055 counts factorizations, strict A045778.

A050320 counts factorizations into squarefree numbers, distinct A050326.

A281113 counts twice-factorizations, strict A296121, see A296118, A296120.

Cf. A000009, A005117, A045782, A050342, A279785, A293243, A293511, A302494, A316439, A358914, A381992, A382201.

Sequence in context: A278136 A085053 A296118 * A296121 A277120 A104725

Adjacent sequences: A383307 A383308 A383309 * A383311 A383312 A383313

Keyword

nonn

Author

Gus Wiseman, Apr 26 2025

Status

approved