A325988 Number of covering (or complete) factorizations of n.

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1

Offset

1,8

Comments

First differs from A072911 at a(64) = 5, A072911(64) = 4.

A covering factorization of n is an orderless factorization of n into factors > 1 such that every divisor of n is the product of some submultiset of the factors.

Links

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

Formula

a(2^n) = A126796(n).

Example

The a(64) = 5 factorizations:
  (2*2*2*2*2*2)
  (2*2*2*2*4)
  (2*2*2*8)
  (2*2*4*4)
  (2*4*8)
The a(96) = 4 factorizations:
  (2*2*2*2*2*3)
  (2*2*2*3*4)
  (2*2*3*8)
  (2*3*4*4)

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

Crossrefs

Cf. A002033, A103295, A126796, A188431.

Cf. A325684, A325781, A325786, A325788, A325986, A325989, A325990.

Sequence in context: A366902 A050361 A072911 * A328856 A053150 A295657

Adjacent sequences: A325985 A325986 A325987 * A325989 A325990 A325991

Keyword

nonn

Author

Gus Wiseman, May 30 2019

Status

approved