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%I #15 Oct 29 2017 02:35:50
%S 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,
%T 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,
%U 4,2,5,1,15,1,2,4,4,2,5,1,11,4,2,1,11,2
%N Number of knapsack factorizations of n.
%C A knapsack factorization is a finite multiset of positive integers greater than one such that every distinct submultiset has a different product.
%C 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
%H Vincenzo Librandi, <a href="/A292886/b292886.txt">Table of n, a(n) for n = 1..2000</a>
%e 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.
%t postfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[postfacs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t Table[Length[Select[postfacs[n],UnsameQ@@Times@@@Union[Subsets[#]]&]],{n,100}]
%Y Cf. A001055, A045778, a(p^n) = A108917(n), A162247, A259936, A275972, A281116.
%K nonn
%O 1,4
%A _Gus Wiseman_, Sep 26 2017
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