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)
84: (4,2,1,1)
210: (4,3,2,1)
490: (4,4,3,1)
525: (4,3,3,2)
2184: (6,4,2,1,1,1)
2340: (6,3,2,2,1,1)
5460: (6,4,3,2,1,1)
9464: (6,6,4,1,1,1)
12012: (6,5,4,2,1,1)
12740: (6,4,4,3,1,1)
12870: (6,5,3,2,2,1)
13650: (6,4,3,3,2,1)
14625: (6,3,3,3,2,2)
19152: (8,4,2,2,1,1,1,1)
For example, the partition (6,6,4,1,1,1) has conjugate (6,3,3,3,2,2), and all four statistics are equal to 3, so 9464 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[1000], Count[primeMS[#], _?EvenQ]==Count[primeMS[#], _?OddQ]==Count[conj[primeMS[#]], _?EvenQ]==Count[conj[primeMS[#]], _?OddQ]&]
CROSSREFS
These partitions are counted by A351978.
There are four individual statistics:
There are six possible pairings of statistics:
There are three possible double-pairings of statistics:
A122111 represents partition conjugation using Heinz numbers.
A195017 = # of even parts - # of odd parts.
A316524 = alternating sum of prime indices.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Mar 14 2022
STATUS
approved