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

1,2

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.

Links

Table of n, a(n) for n=1..18.

Example

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}

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]]=={}]&]

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.

Sequence in context: A216920 A120754 A023547 * A190690 A355509 A084396

Adjacent sequences: A383704 A383705 A383706 * A383708 A383709 A383710

Keyword

nonn,more

Author

Gus Wiseman, May 15 2025

Status

approved