A350946 Heinz numbers of integer partitions with as many even parts as odd parts and as many even conjugate parts as odd conjugate parts.

1, 6, 65, 84, 210, 216, 319, 490, 525, 532, 731, 1254, 1403, 1924, 2184, 2340, 2449, 2470, 3024, 3135, 3325, 3774, 4028, 4141, 4522, 5311, 5460, 7030, 7314, 7315, 7560, 7776, 7942, 8201, 8236, 9048, 9435, 9464, 10659, 10921, 11484, 11914, 12012, 12025, 12740

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)
    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.
		

Crossrefs

For the first condition alone:

- counted by A045931 (strict A239241)

- ordered version (compositions) A098123

- ranked by A325698

- without multiplicity A325700 (counted by A241638)

The second condition alone is ranked by A350848, strict A352129.

These partitions are counted by A351977, strict A352128.

There are four statistics:

- A257991 = # of odd parts, conjugate A344616.

- A257992 = # of even parts, conjugate A350847.

There are four other possible pairings of statistics:

- A349157: # of even parts = # of odd conjugate parts, counted by A277579.

- A350943: # of even conj parts = # of odd parts, strict counted by A352130.

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

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

There are two other possible double-pairings of statistics:

- A350949, counted by A351976.

- A351980, counted by A351981.

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

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.

Cf. A026424, A028260, A130780, A171966, A347450, 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],#==1||Mean[Mod[primeMS[#],2]]== Mean[Mod[conj[primeMS[#]],2]]==1/2&]

Formula

Closed under A122111 (conjugation).

Intersection of A325698 and A350848.

A257992(a(n)) = A257991(a(n)).

A350847(a(n)) = A344616(a(n)).