A383520 Heinz numbers of section-sum partitions with distinct multiplicities (Wilf).

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 20, 23, 25, 27, 28, 29, 31, 32, 37, 40, 41, 43, 44, 45, 47, 49, 50, 52, 53, 56, 59, 61, 64, 67, 68, 71, 73, 75, 76, 79, 80, 81, 83, 88, 89, 92, 97, 98, 99, 101, 103, 104, 107, 109, 112, 113, 116, 117, 121, 124, 125

Offset

1,2

Comments

First differs from A383515 in having 325.

First differs from A383532 in having 325.

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.

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.

Links

Table of n, a(n) for n=1..64.

Example

The terms together with their prime indices begin:
    1: {}
    2: {1}
    3: {2}
    4: {1,1}
    5: {3}
    7: {4}
    8: {1,1,1}
    9: {2,2}
   11: {5}
   13: {6}
   16: {1,1,1,1}
   17: {7}
   19: {8}
   20: {1,1,3}
   23: {9}
   25: {3,3}
   27: {2,2,2}
   28: {1,1,4}
   29: {10}
   31: {11}
   32: {1,1,1,1,1}

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[100], disjointFamilies[conj[prix[#]]]!={}&&UnsameQ@@Last/@FactorInteger[#]&]

Crossrefs

Ranking sequences are shown in parentheses below.

For non Wilf instead of Wilf we have (A383514), counted by A383506.

These partitions are counted by A383519.

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

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

A351592 counts non Wilf Look-and-Say partitions, ranked by (A384006).

A381431 is the section-sum transform.

Cf. A000720, A001223, A048767, A051903, A212166, A320348, A325366, A325368.

Sequence in context: A133811 A316525 A383515 * A383532 A119314 A317923

Adjacent sequences: A383517 A383518 A383519 * A383521 A383522 A383523

Keyword

nonn

Author

Gus Wiseman, May 19 2025

Status

approved