OFFSET
1,5
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
First positions n such that a(n) = 4, 3, 2, 1, 0, -1, -2, -3, -4, together with their prime indices, are:
22: (5,1)
5: (3)
10: (3,1)
2: (1)
1: ()
18: (2,2,1)
9: (2,2)
162: (2,2,2,2,1)
81: (2,2,2,2)
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]}]];
Table[Count[conj[primeMS[n]], _?OddQ]-Count[primeMS[n], _?EvenQ], {n, 100}]
CROSSREFS
Positions of 0's are A349157.
Counting even conjugate parts instead of even parts gives A350941.
The conjugate version is A350942.
The following rank partitions:
A325698: # of even parts = # of odd parts.
A122111 represents conjugation using Heinz numbers.
A316524 = alternating sum of prime indices.
KEYWORD
sign
AUTHOR
Gus Wiseman, Jan 28 2022
STATUS
approved