A292886 Number of knapsack factorizations of n.

1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 4, 1, 2, 2, 4, 1, 4, 1, 4, 2, 2, 1, 7, 2, 2, 3, 4, 1, 5, 1, 6, 2, 2, 2, 8, 1, 2, 2, 7, 1, 5, 1, 4, 4, 2, 1, 11, 2, 4, 2, 4, 1, 7, 2, 7, 2, 2, 1, 11, 1, 2, 4, 7, 2, 5, 1, 4, 2, 5, 1, 15, 1, 2, 4, 4, 2, 5, 1, 11, 4, 2, 1, 11, 2

Offset: 1

Author

Gus Wiseman, Sep 26 2017

Keywords

• nonn

Comments

A knapsack factorization is a finite multiset of positive integers greater than one such that every distinct submultiset has a different product.

The sequence giving the number of factorizations of n is described as "the multiplicative partition function" (see A001055), so knapsack factorizations are a multiplicative generalization of knapsack partitions. - Gus Wiseman, Oct 24 2017

Examples

			The a(36) = 8 factorizations are 2*2*3*3, 2*2*9, 2*18, 3*3*4, 3*12, 4*9, 6*6, 36. The factorization 2*3*6 is not knapsack.
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..2000

Crossrefs

Cf. A001055, A045778, a(p^n) = A108917(n), A162247, A259936, A275972, A281116.

Programs

Mathematica

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