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A326018
Heinz numbers of knapsack partitions such that no addition of one part up to the maximum is knapsack.
6
1925, 12155, 20995, 23375, 37145
OFFSET
1,1
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.
EXAMPLE
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}
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]=={}]&]
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Jun 03 2019
STATUS
approved