A364531 Positive integers with no prime index equal to the sum of prime indices of any nonprime divisor.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 61, 62, 64, 65, 66, 67, 68, 69, 71, 73, 74, 75, 76, 77

Offset: 1

Author

Gus Wiseman, Aug 01 2023

Keywords

• nonn

Comments

First differs from A299702 (knapsack) in having 525: {2,3,3,4}.

First differs from A325778 in lacking 462: {1,2,4,5}.

These are the Heinz numbers of partitions whose parts are disjoint from their own non-singleton subset-sums.

Crossrefs

Partitions of this type are counted by A237667, strict A364349.

The binary version is A364462, complement A364461.

The complement is A364532, counted by A237668.

A000005 counts divisors, nonprime A033273, composite A055212.

A299701 counts distinct subset-sums of prime indices.

A299702 ranks knapsack partitions, counted by A108917, complement A299729.

A363260 counts partitions disjoint from differences, complement A364467.

Cf. A002865, A236912, A237113, A320340, A326083, A363225, A364347.

Programs

Mathematica

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Intersection[prix[#],Total/@Subsets[prix[#],{2,Length[prix[#]]}]]=={}&]