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A350947
Heinz numbers of integer partitions with the same number of even parts, odd parts, even conjugate parts, and odd conjugate parts.
19
1, 6, 84, 210, 490, 525, 2184, 2340, 5460, 9464, 12012, 12740, 12870, 13650, 14625, 19152, 22308, 30030, 34125, 43940, 45144, 55770, 59150, 66066, 70070, 70785, 75075, 79625, 82992, 88920
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
A257992(a(n)) = A257991(a(n)) = A350847(a(n)) = A344616(a(n)).
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:
- A257991 counts odd parts, conjugate A344616.
- A257992 counts even parts, conjugate A350847.
There are six possible pairings of statistics:
- A325698: # of even parts = # of odd parts, counted by A045931.
- A349157: # of even parts = # of odd conjugate parts, counted by A277579.
- A350848: # of even conj parts = # of odd conj parts, counted by A045931.
- A350943: # of even conjugate parts = # of odd parts, counted by A277579.
- A350944: # of odd parts = # of odd conjugate parts, counted by A277103.
- A350945: # of even parts = # of even conjugate parts, counted by A350948.
There are three possible double-pairings of statistics:
- A350946, counted by A351977.
- A350949, counted by A351976.
- A351980, counted by A351981.
A056239 adds up prime indices, counted by A001222, row sums of A112798.
A122111 represents partition conjugation using Heinz numbers.
A195017 = # of even parts - # of odd parts.
A316524 = alternating sum of prime indices.
Sequence in context: A067249 A351980 A351178 * A288321 A155191 A211171
KEYWORD
nonn
AUTHOR
Gus Wiseman, Mar 14 2022
STATUS
approved