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A366529 Heinz numbers of integer partitions of even numbers with at least one even part. 2
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 (list; graph; refs; listen; history; text; internal format)
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
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.
Sequence in context: A085074 A175637 A110404 * A190366 A342699 A284819
KEYWORD
nonn
AUTHOR
Gus Wiseman, Oct 16 2023
STATUS
approved

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Last modified February 22 22:43 EST 2024. Contains 370265 sequences. (Running on oeis4.)