A325990 Numbers with more than one perfect factorization.

8, 24, 27, 32, 40, 54, 56, 72, 88, 96, 104, 108, 120, 125, 128, 135, 136, 152, 160, 168, 184, 189, 200, 216, 224, 232, 243, 248, 250, 256, 264, 270, 280, 288, 296, 297, 312, 328, 343, 344, 351, 352, 360, 375, 376, 378, 384, 392, 408, 416, 424, 432, 440, 456

Offset

1,1

Comments

First differs from A060476 in lacking 1 and having 432.

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..54.

Mathematica

facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];
Select[Range[100], Function[n, Length[Select[facs[n], Sort[Times@@@Union[Subsets[#]]]==Divisors[n]&]]>1]]

Crossrefs

Positions of terms > 1 in A325989.

Cf. A002033, A292886, A325780, A325781, A325782, A325787, A325789, A325988.

Sequence in context: A244030 A375432 A374590 * A060476 A295661 A376142

Adjacent sequences: A325987 A325988 A325989 * A325991 A325992 A325993

Keyword

nonn

Author

Gus Wiseman, May 30 2019

Status

approved