OFFSET
1,25
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.
FORMULA
EXAMPLE
The prime indices of 78 are (6,2,1), with conjugate (3,2,1,1,1,1), so a(78) = 2 - 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]}]];
Table[Count[primeMS[n], _?EvenQ]-Count[conj[primeMS[n]], _?EvenQ], {n, 100}]
CROSSREFS
The version comparing even with odd parts is A195017.
The version comparing even with odd conjugate parts is A350849.
The version comparing even conjugate with odd conjugate parts is A350941.
The version comparing odd with even conjugate parts is A350942.
The version comparing odd with odd conjugate parts is A350951.
There are four individual statistics:
There are five other possible pairings of statistics:
There are three possible double-pairings of statistics:
A116482 counts partitions by number of even parts.
A122111 represents partition conjugation using Heinz numbers.
A316524 gives the alternating sum of prime indices.
KEYWORD
sign
AUTHOR
Gus Wiseman, Mar 14 2022
STATUS
approved