A384390 Heinz numbers of integer partitions with a unique proper way to choose disjoint strict partitions of each part.

5, 7, 21, 22, 26, 33, 35, 39, 102, 114, 130, 154, 165, 170, 190, 195, 231, 238, 255, 285

Offset: 1

Author

Gus Wiseman, Jun 02 2025

Keywords

• nonn

Comments

By "proper" we exclude the case of all singletons, which is disjoint in the strict case.

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.

Examples

			The strict partition (7,2,1) with Heinz number 102 can only be properly refined as ((4,3),(2),(1)), so 102 is in the sequence. The other refinement ((7),(2),(1)) is not proper.
The terms together with their prime indices begin:
    5: {3}
    7: {4}
   21: {2,4}
   22: {1,5}
   26: {1,6}
   33: {2,5}
   35: {3,4}
   39: {2,6}
  102: {1,2,7}
  114: {1,2,8}
  130: {1,3,6}
  154: {1,4,5}
  165: {2,3,5}
  170: {1,3,7}
  190: {1,3,8}
  195: {2,3,6}
  231: {2,4,5}
  238: {1,4,7}
  255: {2,3,7}
  285: {2,3,8}
		

Crossrefs

The non-proper version is A383707, counted by A179009.

Partitions of this type are counted by A384319, non-strict A384323 (ranks A384347).

This is the unique case of A384321, counted by A384317.

This is the case of a unique proper choice in A384322.

The complement is A384349 \/ A384393.

These are positions of 1 in A384389.

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.

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.

A357982 counts strict partitions of each prime index, non-strict A299200.

Cf. A382912, counted by A383710, odd case A383711.

Cf. A382913, counted by A383708, odd case A383533.

Cf. A098859, A122111, A130091, A279375, A279790, A317142, A351201, A381454, A384005, A384320.

Programs

Mathematica

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