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

1,1

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.

Links

Table of n, a(n) for n=1..5.

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]=={}]&]

Crossrefs

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

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

Sequence in context: A135648 A363821 A255867 * A202051 A283949 A133301

Adjacent sequences: A326015 A326016 A326017 * A326019 A326020 A326021

Keyword

nonn,more

Author

Gus Wiseman, Jun 03 2019

Status

approved