A325989 Number of perfect factorizations of n.

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset

1,8

Comments

A perfect factorization of n is an orderless factorization of n into factors > 1 such that every divisor of n is the product of exactly one submultiset of the factors. This is the intersection of covering (or complete) factorizations (A325988) and knapsack factorizations (A292886).

Links

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

Formula

a(2^n) = A002033(n).

Example

The a(216) = 4 perfect factorizations:
  (2*2*2*3*3*3)
  (2*2*2*3*9)
  (2*3*3*3*4)
  (2*3*4*9)

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], Sort[Times@@@Union[Subsets[#]]]==Divisors[n]&]], {n, 100}]

Crossrefs

Positions of terms > 1 are A325990.

Cf. A002033, A103300, A292886, A325685, A325780, A325787, A325789, A325988.

Sequence in context: A359411 A367516 A368979 * A055229 A367698 A367931

Adjacent sequences: A325986 A325987 A325988 * A325990 A325991 A325992

Keyword

nonn

Author

Gus Wiseman, May 30 2019

Status

approved