%I #4 Jan 29 2022 12:49:52

%S 1,2,5,6,8,9,11,14,17,20,21,23,24,26,30,31,32,36,38,39,41,44,47,56,57,

%T 58,59,66,67,68,73,74,75,80,83,84,86,87,92,96,97,102,103,104,106,109,

%U 111,120,122,124,125,127,128,129,137,138,142,144,149,152,156

%N Heinz numbers of integer partitions of which the number of even parts is equal to the number of even conjugate parts.

%C 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.

%F A257992(a(n)) = A350847(a(n)).

%e The terms together with their prime indices begin:

%e 1: ()

%e 2: (1)

%e 5: (3)

%e 6: (2,1)

%e 8: (1,1,1)

%e 9: (2,2)

%e 11: (5)

%e 14: (4,1)

%e 17: (7)

%e 20: (3,1,1)

%e 21: (4,2)

%e 23: (9)

%e 24: (2,1,1,1)

%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];

%t conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];

%t Select[Range[100],Count[conj[primeMS[#]],_?EvenQ]==Count[primeMS[#],_?EvenQ]&]

%Y These partitions are counted by A350948.

%Y These are the positions of 0's in A350950.

%Y A000041 = integer partitions, strict A000009.

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

%Y A122111 = conjugation using Heinz numbers.

%Y A257991 = # of odd parts, conjugate A344616.

%Y A257992 = # of even parts, conjugate A350847.

%Y A316524 = alternating sum of prime indices.

%Y The following rank partitions:

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

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

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

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

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

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

%Y Cf. A000070, A000290, A027187, A027193, A103919, A236559, A344607, A344651, A345196, `A350942, A350950, A350951.

%K nonn

%O 1,2

%A _Gus Wiseman_, Jan 28 2022