A366530 Heinz numbers of integer partitions of even numbers with at least one odd part.

4, 10, 12, 16, 22, 25, 28, 30, 34, 36, 40, 46, 48, 52, 55, 62, 64, 66, 70, 75, 76, 82, 84, 85, 88, 90, 94, 100, 102, 108, 112, 115, 116, 118, 120, 121, 130, 134, 136, 138, 144, 146, 148, 154, 155, 156, 160, 165, 166, 172, 175, 184, 186, 187, 190, 192, 194, 196

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..58.

Example

The terms together with their prime indices are the following. Each multiset has even sum and at least one odd part.
    4: {1,1}
   10: {1,3}
   12: {1,1,2}
   16: {1,1,1,1}
   22: {1,5}
   25: {3,3}
   28: {1,1,4}
   30: {1,2,3}
   34: {1,7}
   36: {1,1,2,2}
   40: {1,1,1,3}
   46: {1,9}
   48: {1,1,1,1,2}
   52: {1,1,6}
   55: {3,5}
   62: {1,11}
   64: {1,1,1,1,1,1}

Mathematica

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

Crossrefs

These partitions are counted by A182616, even bisection of A086543.

Not requiring at least one odd part gives A300061.

Allowing partitions of odd numbers gives A366322.

A031368 lists primes of odd index.

A066207 ranks partitions with all even parts, counted by A035363.

A066208 ranks partitions with all odd parts, counted by A000009.

A112798 list prime indices, sum A056239.

A257991 counts odd prime indices, distinct A324966.

Cf. A000720, A001222, A003963, A047967, A257992, A324929, A358137.

Sequence in context: A380535 A077654 A095275 * A117099 A026180 A109274

Adjacent sequences: A366527 A366528 A366529 * A366531 A366532 A366533

Keyword

nonn

Author

Gus Wiseman, Oct 16 2023

Status

approved