A350848 Heinz numbers of integer partitions for which the number of even conjugate parts is equal to the number of odd conjugate parts.

1, 6, 18, 21, 24, 54, 65, 70, 72, 84, 96, 133, 147, 162, 182, 189, 210, 216, 260, 280, 288, 319, 336, 384, 418, 429, 481, 486, 490, 525, 532, 546, 585, 588, 630, 648, 728, 731, 741, 754, 756, 840, 845, 864, 1007, 1029, 1040, 1120, 1152, 1197, 1254, 1258, 1276

Offset: 1

Author

Gus Wiseman, Jan 27 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)
  18: (2,2,1)
  21: (4,2)
  24: (2,1,1,1)
  54: (2,2,2,1)
  65: (6,3)
  70: (4,3,1)
  72: (2,2,1,1,1)
  84: (4,2,1,1)
  96: (2,1,1,1,1,1)
		

Crossrefs

These partitions are counted by A045931.

The conjugate strict version is counted by A239241.

The conjugate version is A325698.

These are the positions of 0's in A350941.

Adding the conjugate condition gives A350946, all four equal A350947.

A257991 counts odd parts, conjugate A344616.

A257992 counts even parts, conjugate A350847.

A325698: # of even parts = # of odd parts.

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

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

A350943: # of even conjugate parts = # of odd parts, counted by A277579.

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

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

A000041 = integer partitions, strict A000009.

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

A316524 = alternating sum of prime indices, reverse A344616.

Cf. A024619, A026424, A028260, A103919, A130780, A171966, A195017, A241638, A325700, A350849, A350942, A350949.

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

Formula

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

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