login
Number of knapsack factorizations whose factors sum to n.
11

%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