A383518 Heinz numbers of integer partitions that are Look-and-Say and section-sum but not conjugate Wilf partitions.

325, 845, 931, 1625, 2527, 3509, 6253, 6517, 8125, 9251

Offset: 1

Author

Gus Wiseman, May 18 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.

An integer partition is Look-and-Say iff it is possible to choose a disjoint family of strict partitions, one of each of its multiplicities. These are ranked by A351294.

An integer partition is section-sum iff it is possible to choose a disjoint family of strict partitions, one of each of its positive 0-appended differences. These are ranked by A381432.

A integer partition is Wilf iff its multiplicities are all different (ranked by A130091). It is conjugate Wilf iff its positive 0-appended differences are all different (ranked by A383512).

Examples

			The terms together with their prime indices begin:
   325: {3,3,6}
   845: {3,6,6}
   931: {4,4,8}
  1625: {3,3,3,6}
  2527: {4,8,8}
  3509: {5,5,10}
  6253: {6,6,12}
  6517: {4,4,4,8}
  8125: {3,3,3,3,6}
  9251: {5,10,10}
		

Crossrefs

Ranking sequences are shown in parentheses below.

These partitions are counted by A383511.

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, counted by A001222.

A098859 counts Wilf partitions (A130091), conjugate (A383512).

A122111 represents conjugation in terms of Heinz numbers.

A239455 counts Look-and-Say partitions (A351294), complement A351293 (A351295).

A239455 counts section-sum partitions (A381432), complement A351293 (A381433).

A336866 counts non Wilf partitions (A130092), conjugate (A383513).

A381431 is the section-sum transform.

A383508 counts partitions that are both Look-and-Say and section-sum (A383515).

A383509 counts partitions that are Look-and-Say but not section-sum (A383516).

A383509 counts partitions that are not Look-and-Say but are section-sum (A384007).

A383510 counts partitions that are neither Look-and-Say nor section-sum (A383517).

Cf. A047966, A051903, A212166, A238745, A325368, A351592, A383531.

Programs

Mathematica

disjointFamilies[y_]:=Select[Tuples[IntegerPartitions/@Length/@Split[y]],UnsameQ@@Join@@#&];
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]}]];
Select[Range[1000],disjointFamilies[prix[#]]!={}&&disjointFamilies[conj[prix[#]]]!={}&&!UnsameQ@@Length/@Split[conj[prix[#]]]&]