A384010 Heinz numbers of integer partitions such that it is possible to choose a family of disjoint strict partitions, one of each conjugate part.

1, 2, 4, 6, 8, 12, 16, 18, 24, 27, 30, 32, 36, 48, 54, 60, 64, 72, 81, 90, 96, 108, 120, 128, 144, 150, 162, 180, 192

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.

Links

Table of n, a(n) for n=1..29.

Example

The prime indices of 96 are {1,1,1,1,1,2}, conjugate (6,1), disjoint family (4,2,1), so 96 is in the sequence.
The terms together with their prime indices begin:
    1: {}
    2: {1}
    4: {1,1}
    6: {1,2}
    8: {1,1,1}
   12: {1,1,2}
   16: {1,1,1,1}
   18: {1,2,2}
   24: {1,1,1,2}
   27: {2,2,2}
   30: {1,2,3}
   32: {1,1,1,1,1}
   36: {1,1,2,2}
   48: {1,1,1,1,2}
   54: {1,2,2,2}
   60: {1,1,2,3}
   64: {1,1,1,1,1,1}

Mathematica

pof[y_]:=Select[Join@@@Tuples[IntegerPartitions/@y], UnsameQ@@#&];
conj[y_]:=If[Length[y]==0, y, Table[Length[Select[y, #>=k&]], {k, 1, Max[y]}]];
prix[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[100], pof[conj[prix[#]]]!={}&]

Crossrefs

For multiplicities instead of indices we have A382525.

These partitions are counted by A383708, without ones A383533, complement A383711.

These are the positions of positive terms in A384005.

The complement is A384011, conjugate A383710.

A000041 counts integer partitions, strict A000009.

A048767 is the Look-and-Say transform, fixed points A048768, counted by A217605.

A055396 gives least prime index, greatest A061395.

A056239 adds up prime indices, row sums of A112798.

A122111 represent conjugation in terms of Heinz numbers.

A239455 counts Look-and-Say or section-sum partitions, ranks A351294 or A381432.

A351293 counts non-Look-and-Say or non-section-sum partitions, ranks A351295 or A381433.

Cf. A098859, A130091, A279375, A279790, A299200, A357982, A382912, A383706, A383707.

Sequence in context: A060765 A140110 A128397 * A377385 A120383 A324842

Adjacent sequences: A384007 A384008 A384009 * A384011 A384012 A384013

Keyword

nonn,more

Author

Gus Wiseman, May 23 2025

Status

approved