A366529 Heinz numbers of integer partitions of even numbers with at least one even part.

3, 7, 9, 12, 13, 19, 21, 27, 28, 29, 30, 36, 37, 39, 43, 48, 49, 52, 53, 57, 61, 63, 66, 70, 71, 75, 76, 79, 81, 84, 87, 89, 90, 91, 101, 102, 107, 108, 111, 112, 113, 116, 117, 120, 129, 130, 131, 133, 138, 139, 144, 147, 148, 151, 154, 156, 159, 163, 165

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.

Links

Table of n, a(n) for n=1..59.

Example

The terms together with their prime indices begin:
   3: {2}
   7: {4}
   9: {2,2}
  12: {1,1,2}
  13: {6}
  19: {8}
  21: {2,4}
  27: {2,2,2}
  28: {1,1,4}
  29: {10}
  30: {1,2,3}
  36: {1,1,2,2}
  37: {12}
  39: {2,6}
  43: {14}
  48: {1,1,1,1,2}

Mathematica

prix[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[100], EvenQ[Total[prix[#]]]&&Or@@EvenQ/@prix[#]&]

Crossrefs

The complement is counted by A047967.

For all even parts we have A066207, counted by A035363, odd A066208.

Not requiring an even part gives A300061.

For odd instead of even we have A300063.

Not requiring even sum gives A324929.

Partitions of this type are counted by A366527.

A112798 list prime indices, sum A056239.

A257991 counts odd prime indices, distinct A324966.

A257992 counts even prime indices, distinct A324967.

Cf. A000720, A001222, A003963, A033844, A086543, A324927, A358137, A366530.

Sequence in context: A085074 A175637 A110404 * A190366 A342699 A284819

Adjacent sequences: A366526 A366527 A366528 * A366530 A366531 A366532

Keyword

nonn

Author

Gus Wiseman, Oct 16 2023

Status

approved