A350944 Heinz numbers of integer partitions of which the number of odd parts is equal to the number of odd conjugate parts.

1, 2, 6, 9, 10, 12, 15, 18, 20, 30, 35, 49, 54, 55, 56, 70, 75, 77, 81, 84, 88, 90, 98, 108, 110, 112, 125, 132, 135, 143, 154, 162, 168, 169, 176, 180, 187, 210, 221, 260, 264, 270, 286, 294, 315, 323, 330, 338, 340, 350, 361, 363, 364, 374, 391, 416, 420

Offset: 1

Author

Gus Wiseman, Jan 28 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: ()
   2: (1)
   6: (2,1)
   9: (2,2)
  10: (3,1)
  12: (2,1,1)
  15: (3,2)
  18: (2,2,1)
  20: (3,1,1)
  30: (3,2,1)
  35: (4,3)
  49: (4,4)
  54: (2,2,2,1)
		

Crossrefs

These partitions are counted by A277103.

The even rank case is A345196.

The conjugate version is A350943, counted by A277579.

These are the positions of 0's in A350951, even A350950.

A000041 = integer partitions, strict A000009.

A056239 adds up prime indices, counted by A001222, row sums of A112798.

A122111 = conjugation using Heinz numbers.

A257991 = # of odd parts, conjugate A344616.

A257992 = # of even parts, conjugate A350847.

A316524 = alternating sum of prime indices.

The following rank partitions:

A325040: product = product of conjugate, counted by A325039.

A325698: # of even parts = # of odd parts, counted by A045931.

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

A350848: # even conj parts = # odd conj parts, counted by A045931.

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

Cf. A000070, A000700, A027187, A027193, A103919, A236559, A350942.

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

Formula

A257991(a(n)) = A344616(a(n)).