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.
FORMULA
EXAMPLE
The terms together with their prime indices begin:
1: ()
6: (2,1)
65: (6,3)
84: (4,2,1,1)
210: (4,3,2,1)
216: (2,2,2,1,1,1)
319: (10,5)
490: (4,4,3,1)
525: (4,3,3,2)
532: (8,4,1,1)
731: (14,7)
1254: (8,5,2,1)
1403: (18,9)
1924: (12,6,1,1)
2184: (6,4,2,1,1,1)
2340: (6,3,2,2,1,1)
2449: (22,11)
2470: (8,6,3,1)
For example, the prime indices of 532 are (8,4,1,1), even/odd counts 2/2, and the prime indices of the conjugate 3024 are (4,2,2,2,1,1,1,1), with even/odd counts 4/4; so 532 belongs to 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], #==1||Mean[Mod[primeMS[#], 2]]== Mean[Mod[conj[primeMS[#]], 2]]==1/2&]
CROSSREFS
For the first condition alone:
- ordered version (compositions) A098123
- ranked by A325698
There are four statistics:
There are four other possible pairings of statistics:
There are two other 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