A351980 Heinz numbers of integer partitions with as many even parts as odd conjugate parts and as many odd parts as even conjugate parts.

1, 6, 84, 126, 140, 210, 490, 525, 686, 875, 1404, 1456, 2106, 2184, 2288, 2340, 3432, 3510, 5460, 6760, 7644, 8190, 8580, 8775, 9100, 9464, 11466, 12012, 12740, 12870, 13650, 14300, 14625, 15808, 18018, 18468, 19110, 19152, 20020, 20672, 21450, 22308, 23712

Offset: 1

Author

Gus Wiseman, Mar 14 2022

Keywords

• nonn

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.

Examples

			The terms together with their prime indices begin:
     1: ()
     6: (2,1)
    84: (4,2,1,1)
   126: (4,2,2,1)
   140: (4,3,1,1)
   210: (4,3,2,1)
   490: (4,4,3,1)
   525: (4,3,3,2)
   686: (4,4,4,1)
   875: (4,3,3,3)
  1404: (6,2,2,2,1,1)
  1456: (6,4,1,1,1,1)
  2106: (6,2,2,2,2,1)
  2184: (6,4,2,1,1,1)
  2288: (6,5,1,1,1,1)
  2340: (6,3,2,2,1,1)
		

Crossrefs

The first condition alone is A349157, counted by A277579.

The second condition alone is A350943, counted by A277579.

There are two other possible double-pairings of statistics:

- A350946, counted by A351977.

- A350949, counted by A351976.

The case of all four statistics equal is A350947, counted by A351978.

These partitions are counted by A351981.

Partitions with as many even as odd parts:

- counted by A045931

- strict case counted by A239241

- ranked by A325698

- conjugate ranked by A350848

- strict conjugate case counted by A352129

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.

A257991 counts odd parts, conjugate A344616.

A257992 counts even parts, conjugate A350847.

A316524 = alternating sum of prime indices.

A350944: # of odd parts = # of odd conjugate parts, counted by A277103.

A350945: # of even parts = # of even conjugate parts, counted by A350948.

Cf. A026424, A028260, A098123, A130780, A171966, A241638, A325700, A350841, A350849, A350941, A350942, A350950, A350951.

Programs

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[conj[primeMS[#]],?OddQ]&&Count[primeMS[#],?OddQ]==Count[conj[primeMS[#]],?EvenQ]&]

Formula

Closed under A122111 (conjugation).

Intersection of A349157 and A350943.

A257992(a(n)) = A344616(a(n)).

A257991(a(n)) = A350847(a(n)).