A349157 Heinz numbers of integer partitions where the number of even parts is equal to the number of odd conjugate parts.

1, 4, 6, 15, 16, 21, 24, 25, 35, 60, 64, 77, 84, 90, 91, 96, 100, 121, 126, 140, 143, 150, 210, 221, 240, 247, 256, 289, 297, 308, 323, 336, 351, 360, 364, 375, 384, 400, 437, 462, 484, 490, 495, 504, 525, 529, 546, 551, 560, 572, 585, 600, 625, 667, 686, 726

Offset: 1

Author

Gus Wiseman, Jan 21 2022

Keywords

• nonn

Comments

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are numbers with the same number of even prime indices as odd conjugate prime indices.

These are also partitions for which the number of even parts is equal to the positive alternating sum of the parts.

Examples

			The terms and their prime indices begin:
    1: ()
    4: (1,1)
    6: (2,1)
   15: (3,2)
   16: (1,1,1,1)
   21: (4,2)
   24: (2,1,1,1)
   25: (3,3)
   35: (4,3)
   60: (3,2,1,1)
   64: (1,1,1,1,1,1)
   77: (5,4)
   84: (4,2,1,1)
   90: (3,2,2,1)
   91: (6,4)
   96: (2,1,1,1,1,1)
		

Crossrefs

A subset of A028260 (even bigomega), counted by A027187.

These partitions are counted by A277579.

This is the half-conjugate version of A325698, counted by A045931.

A000041 counts partitions, strict A000009.

A047993 counts balanced partitions, ranked by A106529.

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

A100824 counts partitions with at most one odd part, ranked by A349150.

A108950/A108949 count partitions with more odd/even parts.

A122111 represents conjugation using Heinz numbers.

A130780/A171966 count partitions with more or equal odd/even parts.

A257991/A257992 count odd/even prime indices.

A316524 gives the alternating sum of prime indices (reverse: A344616).

Cf. A000700, A000712, A035363, A066207, A066208, A097613, A215366, A239241, A240009, A241638, A316523, A325700, A340604.

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

Formula

A257992(a(n)) = A257991(A122111(a(n))).