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A325989 Number of perfect factorizations of n. 2
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 (list; graph; refs; listen; history; text; internal format)
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: A295658 A307427 A318672 * A055229 A270419 A275216

Adjacent sequences:  A325986 A325987 A325988 * A325990 A325991 A325992

KEYWORD

nonn

AUTHOR

Gus Wiseman, May 30 2019

STATUS

approved

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Last modified January 19 00:40 EST 2020. Contains 331030 sequences. (Running on oeis4.)