A350943 Heinz numbers of integer partitions of which the number of even conjugate parts is equal to the number of odd parts.

1, 3, 6, 7, 13, 14, 18, 19, 26, 27, 29, 36, 37, 38, 42, 43, 53, 54, 58, 61, 63, 70, 71, 74, 78, 79, 84, 86, 89, 101, 105, 106, 107, 113, 114, 117, 122, 126, 130, 131, 139, 140, 142, 151, 156, 158, 162, 163, 171, 173, 174, 178, 181, 190, 193, 195, 199, 202, 210

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.

Links

Table of n, a(n) for n=1..59.

Formula

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

Example

The terms together with their prime indices begin:
   1: ()
   3: (2)
   6: (2,1)
   7: (4)
  13: (6)
  14: (4,1)
  18: (2,2,1)
  19: (8)
  26: (6,1)
  27: (2,2,2)
  29: (10)
  36: (2,2,1,1)
  37: (12)
  38: (8,1)
  42: (4,2,1)
For example, the partition (6,3,2) has conjugate (3,3,2,1,1,1) and 1 = 1 so 195 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[100], Count[primeMS[#], _?OddQ]==Count[conj[primeMS[#]], _?EvenQ]&]

Crossrefs

These partitions are counted by A277579.

The conjugate version is A349157, also counted by A277579.

These are the positions of 0's in A350942.

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.

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

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

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

Cf. A000070, A000290, A027187, A027193, A103919, A236559, A344607, A344651, A345196, A350950, A350951.

Sequence in context: A280873 A293437 A359756 * A282354 A088146 A176301

Adjacent sequences: A350940 A350941 A350942 * A350944 A350945 A350946

Keyword

nonn

Author

Gus Wiseman, Jan 28 2022

Status

approved