

A325990


Numbers with more than one perfect factorization.


2



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
(list;
graph;
refs;
listen;
history;
text;
internal format)



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: A254448 A029607 A244030 * A060476 A295661 A240111
Adjacent sequences: A325987 A325988 A325989 * A325991 A325992 A325993


KEYWORD

nonn


AUTHOR

Gus Wiseman, May 30 2019


STATUS

approved



