A326018 Heinz numbers of knapsack partitions such that no addition of one part up to the maximum is knapsack.

1925, 12155, 20995, 23375, 37145

Offset: 1

Author

Gus Wiseman, Jun 03 2019

Keywords

• nonn
• more

Comments

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

An integer partition is knapsack if every submultiset has a different sum.

The enumeration of these partitions by sum is given by A326016.

Examples

			The sequence of terms together with their prime indices begins:
   1925: {3,3,4,5}
  12155: {3,5,6,7}
  20995: {3,6,7,8}
  23375: {3,3,3,5,7}
  37145: {3,7,8,9}
		

Crossrefs

Cf. A002033, A108917, A275972, A299702, A299729, A304793.

Cf. A325780, A325782, A325857, A325862, A325878, A325880, A326015, A326016.

Programs

Mathematica

ksQ[y_]:=UnsameQ@@Total/@Union[Subsets[y]];
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]=={}]&]