OFFSET
1,2
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.
EXAMPLE
The terms together with their prime indices begin:
1: ()
6: (2,1)
18: (2,2,1)
21: (4,2)
24: (2,1,1,1)
54: (2,2,2,1)
65: (6,3)
70: (4,3,1)
72: (2,2,1,1,1)
84: (4,2,1,1)
96: (2,1,1,1,1,1)
MATHEMATICA
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
conj[y_]:=If[Length[y]==0, y, Table[Length[Select[y, #>=k&]], {k, 1, Max[y]}]];
Select[Range[100], Count[conj[primeMS[#]], _?EvenQ]==Count[conj[primeMS[#]], _?OddQ]&]
CROSSREFS
These partitions are counted by A045931.
The conjugate strict version is counted by A239241.
The conjugate version is A325698.
These are the positions of 0's in A350941.
A325698: # of even parts = # of odd parts.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jan 27 2022
STATUS
approved