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Numbers with more than one perfect factorization.
4

%I #5 May 31 2019 05:34:38

%S 8,24,27,32,40,54,56,72,88,96,104,108,120,125,128,135,136,152,160,168,

%T 184,189,200,216,224,232,243,248,250,256,264,270,280,288,296,297,312,

%U 328,343,344,351,352,360,375,376,378,384,392,408,416,424,432,440,456

%N Numbers with more than one perfect factorization.

%C First differs from A060476 in lacking 1 and having 432.

%C 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).

%t facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];

%t Select[Range[100],Function[n,Length[Select[facs[n],Sort[Times@@@Union[Subsets[#]]]==Divisors[n]&]]>1]]

%Y Positions of terms > 1 in A325989.

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

%K nonn

%O 1,1

%A _Gus Wiseman_, May 30 2019