A327399 Number of factorizations of n that are constant or whose distinct factors are pairwise coprime.

1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 3, 1, 2, 2, 3, 1, 3, 1, 3, 2, 2, 1, 3, 2, 2, 2, 3, 1, 5, 1, 2, 2, 2, 2, 6, 1, 2, 2, 3, 1, 5, 1, 3, 3, 2, 1, 4, 2, 3, 2, 3, 1, 3, 2, 3, 2, 2, 1, 7, 1, 2, 3, 4, 2, 5, 1, 3, 2, 5, 1, 5, 1, 2, 3, 3, 2, 5, 1, 4, 3, 2, 1, 7, 2, 2, 2

Offset

1,4

Comments

First differs from A327400 at A327400(24) = 4, a(24) = 3.

Links

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

Gus Wiseman, Sequences counting and encoding certain classes of multisets

Formula

a(n) = A327695(n) + A089723(n).

Example

The a(90) = 7 factorizations together with the corresponding multiset partitions of {1,2,2,3}:
  (2*3*3*5)  {{1},{2},{2},{3}}
  (2*5*9)    {{1},{3},{2,2}}
  (2*45)     {{1},{2,2,3}}
  (3*3*10)   {{2},{2},{1,3}}
  (5*18)     {{3},{1,2,2}}
  (9*10)     {{2,2},{1,3}}
  (90)       {{1,2,2,3}}

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[#]]==1||CoprimeQ@@Union[#]&]], {n, 100}]

Crossrefs

Constant factorizations are A089723.

Partitions whose distinct parts are pairwise coprime are A304709.

Factorizations that are constant or relatively prime are A327400.

See link for additional cross-references.

A007359, A050320, A051424, A281116, A302569, A302696, A304711.

Sequence in context: A106490 A349281 A345222 * A122375 A038548 A320732

Adjacent sequences: A327396 A327397 A327398 * A327400 A327401 A327402

Keyword

nonn

Author

Gus Wiseman, Sep 22 2019

Status

approved