OFFSET
1,8
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) = 6, 5, 4, 3, 2, 1, 0, -1, -2, -3, -4, -5, -6, together with their prime indices, are:
192: (2,1,1,1,1,1,1)
32: (1,1,1,1,1)
48: (2,1,1,1,1)
8: (1,1,1)
12: (2,1,1)
2: (1)
1: ()
15: (3,2)
9: (2,2)
77: (5,4)
49: (4,4)
221: (7,6)
169: (6,6)
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[primeMS[n], _?OddQ]-Count[conj[primeMS[n]], _?EvenQ], {n, 100}]
CROSSREFS
The conjugate version is A350849.
Positions of 0's are A350943.
A122111 represents conjugation using Heinz numbers.
A316524 = alternating sum of prime indices.
The following rank partitions:
A325698: # of even parts = # of odd parts.
KEYWORD
sign
AUTHOR
Gus Wiseman, Jan 28 2022
STATUS
approved