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%I #5 Jan 29 2022 12:49:45
%S 1,2,6,9,10,12,15,18,20,30,35,49,54,55,56,70,75,77,81,84,88,90,98,108,
%T 110,112,125,132,135,143,154,162,168,169,176,180,187,210,221,260,264,
%U 270,286,294,315,323,330,338,340,350,361,363,364,374,391,416,420
%N Heinz numbers of integer partitions of which the number of odd parts is equal to the number of odd 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 A257991(a(n)) = A344616(a(n)).
%e The terms together with their prime indices begin:
%e 1: ()
%e 2: (1)
%e 6: (2,1)
%e 9: (2,2)
%e 10: (3,1)
%e 12: (2,1,1)
%e 15: (3,2)
%e 18: (2,2,1)
%e 20: (3,1,1)
%e 30: (3,2,1)
%e 35: (4,3)
%e 49: (4,4)
%e 54: (2,2,2,1)
%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[#]],_?OddQ]==Count[primeMS[#],_?OddQ]&]
%Y These partitions are counted by A277103.
%Y The even rank case is A345196.
%Y The conjugate version is A350943, counted by A277579.
%Y These are the positions of 0's in A350951, even 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: # even conj parts = # odd conj parts, counted by A045931.
%Y A350945: # of even parts = # of even conjugate parts, counted by A350948.
%Y Cf. A000070, A000700, A027187, A027193, A103919, A236559, A350942.
%K nonn
%O 1,2
%A _Gus Wiseman_, Jan 28 2022
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