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: ()
3: (2)
6: (2,1)
7: (4)
13: (6)
14: (4,1)
18: (2,2,1)
19: (8)
26: (6,1)
27: (2,2,2)
29: (10)
36: (2,2,1,1)
37: (12)
38: (8,1)
42: (4,2,1)
For example, the partition (6,3,2) has conjugate (3,3,2,1,1,1) and 1 = 1 so 195 is in the sequence.
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[primeMS[#], _?OddQ]==Count[conj[primeMS[#]], _?EvenQ]&]
CROSSREFS
These partitions are counted by A277579.
These are the positions of 0's in A350942.
A122111 = conjugation using Heinz numbers.
A316524 = alternating sum of prime indices.
The following rank partitions:
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jan 28 2022
STATUS
approved