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%I #7 Jun 04 2019 08:37:03
%S 0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,1,1,0,3,0,0,1,1,0,8,0,
%T 8,4,3,0,11,5,3,4,5,0,30,2,9,9,20,3,37,6,18,16,37,20,71,12,37,40
%N Number of knapsack partitions of n such that no addition of one part equal to an existing part is knapsack.
%C An integer partition is knapsack if every distinct submultiset has a different sum.
%e The partition (10,8,6,6) is counted under a(30) because (10,10,8,6,6), (10,8,8,6,6), and (10,8,6,6,6) are not knapsack.
%t sums[ptn_]:=sums[ptn]=If[Length[ptn]==1,ptn,Union@@(Join[sums[#],sums[#]+Total[ptn]-Total[#]]&/@Union[Table[Delete[ptn,i],{i,Length[ptn]}]])];
%t ksQ[y_]:=Length[sums[Sort[y]]]==Times@@(Length/@Split[Sort[y]]+1)-1;
%t maxks[n_]:=Select[IntegerPartitions[n],ksQ[#]&&Select[Table[Sort[Append[#,i]],{i,Union[#]}],ksQ]=={}&];
%t Table[Length[maxks[n]],{n,30}]
%Y Cf. A002033, A108917, A275972, A276024, A299702.
%Y Cf. A325857, A325862, A325863, A325864, A325865, A326015, A326016, A326018.
%K nonn,more
%O 1,21
%A _Gus Wiseman_, Jun 03 2019
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