A383707 Heinz numbers of maximally refined strict integer partitions.

1, 2, 3, 6, 10, 14, 15, 30, 42, 66, 70, 78, 105, 110, 182, 210, 330, 390

Offset: 1

Author

Gus Wiseman, May 15 2025

Keywords

• nonn
• more

Comments

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

Also squarefree numbers such that every strict partition of a prime index contains a prime index.

Also squarefree numbers such that no prime index is a sum of distinct non prime indices.

Examples

			The terms together with their prime indices begin:
    1: {}
    2: {1}
    3: {2}
    6: {1,2}
   10: {1,3}
   14: {1,4}
   15: {2,3}
   30: {1,2,3}
   42: {1,2,4}
   66: {1,2,5}
   70: {1,3,4}
   78: {1,2,6}
  105: {2,3,4}
  110: {1,3,5}
  182: {1,4,6}
  210: {1,2,3,4}
  330: {1,2,3,5}
  390: {1,2,3,6}
		

Crossrefs

Partitions of this type are counted by A179009.

Appears to be positions of 1 in A383706.

For distinct prime indices see A384320.

The proper version appears to be A384390.

The conjugate version is A384723.

A055396 gives least prime index, greatest A061395.

A056239 adds up prime indices, row sums of A112798.

Cf. A048767, A130091, A299200, A351294, A381432, A351295, A357982, A381454, A382525, A384321, A384349, A384389.

Programs

Mathematica

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
nonsets[y_]:=If[Length[y]==0,{},Rest[Subsets[Complement[Range[Max@@y],y]]]];
Select[Range[30],SquareFreeQ[#]&&With[{y=prix[#]},Intersection[y,Total/@nonsets[y]]=={}]&]