A383531 Heinz numbers of integer partitions that do not have distinct multiplicities (Wilf) or distinct nonzero 0-appended differences (conjugate Wilf).

6, 21, 30, 36, 42, 60, 65, 66, 70, 78, 84, 90, 102, 105, 110, 114, 120, 126, 132, 133, 138, 140, 150, 154, 156, 165, 168, 174, 180, 186, 198, 204, 210, 216, 220, 222, 228, 231, 234, 238, 240, 246, 252, 258, 264, 270, 273, 276, 280, 282, 286, 294, 300, 306, 308

Offset

1,1

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 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..55.

Formula

Equals A130092 /\ A383513.

Example

The terms together with their prime indices begin:
    6: {1,2}
   21: {2,4}
   30: {1,2,3}
   36: {1,1,2,2}
   42: {1,2,4}
   60: {1,1,2,3}
   65: {3,6}
   66: {1,2,5}
   70: {1,3,4}
   78: {1,2,6}
   84: {1,1,2,4}
   90: {1,2,2,3}
  102: {1,2,7}
  105: {2,3,4}
  110: {1,3,5}
  114: {1,2,8}
  120: {1,1,1,2,3}

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]}]];
Select[Range[100], !UnsameQ@@Length/@Split[prix[#]] && !UnsameQ@@Length/@Split[conj[prix[#]]]&]

Crossrefs

These partitions are counted by A383530.

Negating both sides gives A383532, counted by A383507.

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.

A098859 counts Wilf partitions, ranks A130091, conjugate A383512.

A122111 represents conjugation in terms of Heinz numbers.

A325324 counts integer partitions with distinct 0-appended differences, ranks A325367.

A336866 counts non Wilf partitions, ranks A130092, conjugate A383513.

A383709 counts Wilf partitions with distinct 0-appended differences, ranks A383712.

Cf. A001223, A047966, A181819, A238745, A320348, A325325, A325349, A325366, A325368, A325388, A381431, A383506.

Sequence in context: A246544 A034897 A347875 * A383517 A287165 A280296

Adjacent sequences: A383528 A383529 A383530 * A383532 A383533 A383534

Keyword

nonn

Author

Gus Wiseman, May 15 2025

Status

approved