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%I #4 Jun 04 2019 08:36:18
%S 1925,12155,20995,23375,37145
%N Heinz numbers of knapsack partitions such that no addition of one part up to the maximum is knapsack.
%C The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
%C An integer partition is knapsack if every submultiset has a different sum.
%C The enumeration of these partitions by sum is given by A326016.
%e The sequence of terms together with their prime indices begins:
%e 1925: {3,3,4,5}
%e 12155: {3,5,6,7}
%e 20995: {3,6,7,8}
%e 23375: {3,3,3,5,7}
%e 37145: {3,7,8,9}
%t ksQ[y_]:=UnsameQ@@Total/@Union[Subsets[y]];
%t Select[Range[2,200],With[{phm=If[#==1,{},Flatten[Cases[FactorInteger[#],{p_,k_}:>Table[PrimePi[p],{k}]]]]},ksQ[phm]&&Select[Table[Sort[Append[phm,i]],{i,Max@@phm}],ksQ]=={}]&]
%Y Cf. A002033, A108917, A275972, A299702, A299729, A304793.
%Y Cf. A325780, A325782, A325857, A325862, A325878, A325880, A326015, A326016.
%K nonn,more
%O 1,1
%A _Gus Wiseman_, Jun 03 2019
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