A383513 Heinz numbers of non conjugate Wilf partitions.

6, 12, 18, 21, 24, 30, 36, 42, 48, 54, 60, 63, 65, 66, 70, 72, 78, 84, 90, 96, 102, 105, 108, 110, 114, 120, 126, 132, 133, 138, 140, 144, 147, 150, 154, 156, 162, 165, 168, 174, 180, 186, 189, 192, 198, 204, 210, 216, 220, 222, 228, 231, 234, 238, 240, 246

Offset

1,1

Comments

First differs from A381433 in having 65.

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

Links

Table of n, a(n) for n=1..56.

Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Example

The terms together with their prime indices begin:
    6: {1,2}
   12: {1,1,2}
   18: {1,2,2}
   21: {2,4}
   24: {1,1,1,2}
   30: {1,2,3}
   36: {1,1,2,2}
   42: {1,2,4}
   48: {1,1,1,1,2}
   54: {1,2,2,2}
   60: {1,1,2,3}
   63: {2,2,4}
   65: {3,6}
   66: {1,2,5}
   70: {1,3,4}
   72: {1,1,1,2,2}
   78: {1,2,6}
   84: {1,1,2,4}
   90: {1,2,2,3}
   96: {1,1,1,1,1,2}

Mathematica

prix[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[100], !UnsameQ@@DeleteCases[Differences[Prepend[prix[#], 0]], 0]&]

Crossrefs

Partitions of this type are counted by A336866.

The conjugate version is A130092, complement A130091.

Including differences of 0 gives complement of A325367, counted by A325324.

The strict case is the complement of A325388, counted by A320348.

The complement is A383512, counted by A098859.

Also forbidding distinct multiplicities gives A383531, counted by A383530.

These are positions of non-strict rows in A383534, or nonsquarefree numbers in A383535.

A000040 lists the primes, differences A001223.

A048767 is the Look-and-Say transform, union A351294, complement A351295.

A055396 gives least prime index, greatest A061395.

A056239 adds up prime indices, row sums of A112798, counted by A001222.

A122111 represents conjugation in terms of Heinz numbers.

A239455 counts Look-and-Say partitions, complement A351293.

A383507 counts partitions that are Wilf and conjugate Wilf, ranks A383532.

A383709 counts Wilf partitions with distinct augmented differences, ranks A383712.

Cf. A000720, A005117, A048768, A238745, A325325, A325351, A325355, A325366, A325368, A381431, A383506.

Sequence in context: A138939 A221220 A046411 * A364348 A364537 A350845

Adjacent sequences: A383510 A383511 A383512 * A383514 A383515 A383516

Keyword

nonn

Author

Gus Wiseman, May 13 2025

Status

approved