A384723 Heinz numbers of conjugates of maximally refined strict integer partitions.

1, 2, 4, 6, 12, 18, 24, 30, 60, 90, 120, 150, 180, 210, 240, 420, 540, 630, 840, 1050, 1260, 1470, 1680, 1890, 2100, 2310, 2520, 3360, 4620, 6300, 6930, 7560, 9240

Offset: 1

Author

Gus Wiseman, Jun 09 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.

Given a partition, the following are equivalent:

1) The distinct parts are maximally refined.

2) Every strict partition of a part contains a part. In other words, if y is the set of parts and z is any strict partition of any element of y, then z must contain at least one element from y.

3) No part is a sum of distinct non-parts.

Examples

			The terms together with their prime indices begin:
     1: {}
     2: {1}
     4: {1,1}
     6: {1,2}
    12: {1,1,2}
    18: {1,2,2}
    24: {1,1,1,2}
    30: {1,2,3}
    60: {1,1,2,3}
    90: {1,2,2,3}
   120: {1,1,1,2,3}
   150: {1,2,3,3}
   180: {1,1,2,2,3}
   210: {1,2,3,4}
   240: {1,1,1,1,2,3}
   420: {1,1,2,3,4}
   540: {1,1,2,2,2,3}
   630: {1,2,2,3,4}
   840: {1,1,1,2,3,4}
		

Crossrefs

Partitions of this type are counted by A179009.

The conjugate version is A383707, proper A384390.

Appears to be the positions of 1 in A384005 (conjugate A383706).

For at least one instead of exactly one choice we appear to have A384010.

A055396 gives least prime index, greatest A061395.

A056239 adds up prime indices, row sums of A112798.

A122111 represents 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.

A357982 counts strict partitions of prime indices, non-strict A299200.

Cf. A003963, A048767, A130091, A382525, A384317, A384318, A384320, A384347, A384349, A384394.

Programs

Mathematica

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
nonsets[y_]:=If[Length[y]==0,{},Rest[Subsets[Complement[Range[Max@@y],y]]]];
Select[Range[100],With[{y=conj[prix[#]]},UnsameQ@@y&&Intersection[y,Total/@nonsets[y]]=={}]&]