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%I #19 Oct 29 2017 21:30:13
%S 1,1,1,2,2,4,4,6,8,11,12,19,21,27,34,45,51,69,77,100,117,146
%N Number of knapsack factorizations whose factors sum to n.
%C A knapsack factorization is a finite multiset of positive integers greater than one such that every distinct submultiset has a different product.
%e The a(12) = 19 partitions are:
%e (12),
%e (10 2), (9 3), (8 4), (7 5), (6 6),
%e (8 2 2), (7 3 2), (6 4 2), (6 3 3), (5 5 2), (5 4 3), (4 4 4),
%e (6 2 2 2), (5 3 2 2), (4 3 3 2), (3 3 3 3),
%e (3 3 2 2 2),
%e (2 2 2 2 2 2).
%t nn=22;
%t apsQ[y_]:=UnsameQ@@Times@@@Union[Rest@Subsets[y]];
%t Table[Length@Select[IntegerPartitions[n],apsQ],{n,nn}]
%Y Cf. A000041, A001055, A002033, A002865, A108917, A126796, A275972, A281116, A292886, A294150.
%K nonn,more
%O 1,4
%A _Gus Wiseman_, Oct 23 2017
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