A366532 Heinz numbers of integer partitions with at least one even and odd part.

6, 12, 14, 15, 18, 24, 26, 28, 30, 33, 35, 36, 38, 42, 45, 48, 51, 52, 54, 56, 58, 60, 65, 66, 69, 70, 72, 74, 75, 76, 77, 78, 84, 86, 90, 93, 95, 96, 98, 99, 102, 104, 105, 106, 108, 112, 114, 116, 119, 120, 122, 123, 126, 130, 132, 135, 138, 140, 141, 142

Offset: 1

Author

Gus Wiseman, Oct 16 2023

Keywords

• nonn

Comments

These partitions are counted by A006477.

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 begin:
    6: {1,2}
   12: {1,1,2}
   14: {1,4}
   15: {2,3}
   18: {1,2,2}
   24: {1,1,1,2}
   26: {1,6}
   28: {1,1,4}
   30: {1,2,3}
   33: {2,5}
   35: {3,4}
   36: {1,1,2,2}
   38: {1,8}
   42: {1,2,4}
   45: {2,2,3}
   48: {1,1,1,1,2}
		

Crossrefs

These partitions are counted by A006477.

Just even: A324929, counted by A047967.

Just odd: A366322, counted by A086543 (even bisection of A182616).

A031368 lists primes of odd index, even A031215.

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

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

A112798 lists prime indices, sum A056239.

A257991 counts odd prime indices, distinct A324966.

A257992 counts even prime indices, distinct A324967.

Cf. A000720, A001222, A003963, A026804, A244991, A257992.

Programs

Mathematica

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

Formula

Intersection of A324929 and A366322.