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

Author

Gus Wiseman, Oct 16 2023

Keywords

• nonn

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.

Examples

			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}
		

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.

Programs

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[#]&]