A384347 Heinz numbers of integer partitions with exactly two possible ways to choose disjoint strict partitions of each part.

5, 7, 21, 22, 25, 26, 33, 35, 39, 49, 102, 114, 130, 147, 154, 165, 170, 175, 190, 195, 231, 238, 242, 255, 275, 285

Offset: 1

Author

Gus Wiseman, May 27 2025

Keywords

• nonn
• more

Comments

Positions of 2 in A383706.

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 prime indices of 275 are {3,3,5}, with two ways to choose disjoint strict partitions of each part: ((3),(2,1),(5)) and ((2,1),(3),(5)). Hence 275 is in the sequence.
The terms together with their prime indices begin:
    5: {3}
    7: {4}
   21: {2,4}
   22: {1,5}
   25: {3,3}
   26: {1,6}
   33: {2,5}
   35: {3,4}
   39: {2,6}
   49: {4,4}
  102: {1,2,7}
  114: {1,2,8}
  130: {1,3,6}
  147: {2,4,4}
  154: {1,4,5}
  165: {2,3,5}
		

Crossrefs

The case of no choices is A382912, counted by A383710, odd case A383711.

These are positions of 2 in A383706.

The case of no proper choices is A383707, counted by A179009.

The case of some proper choice is A384321, strict A384322, count A384317, strict A384318.

These partitions are counted by A384323, strict A384319.

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 prime indices, non-strict A299200.

Cf. A048767, A382525, A382771, A382857, A383533, A383708, A384320.

Programs

Mathematica

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
pof[y_]:=Select[Join@@@Tuples[IntegerPartitions/@y],UnsameQ@@#&];
Select[Range[100],Length[pof[prix[#]]]==2&]