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%I #6 Jan 27 2022 20:47:11
%S 1,6,18,21,24,54,65,70,72,84,96,133,147,162,182,189,210,216,260,280,
%T 288,319,336,384,418,429,481,486,490,525,532,546,585,588,630,648,728,
%U 731,741,754,756,840,845,864,1007,1029,1040,1120,1152,1197,1254,1258,1276
%N Heinz numbers of integer partitions for which the number of even conjugate 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 A344616(a(n)) = A350847(a(n)).
%F A257991(A122111(a(n))) = A257992(A122111(a(n))).
%e The terms together with their prime indices begin:
%e 1: ()
%e 6: (2,1)
%e 18: (2,2,1)
%e 21: (4,2)
%e 24: (2,1,1,1)
%e 54: (2,2,2,1)
%e 65: (6,3)
%e 70: (4,3,1)
%e 72: (2,2,1,1,1)
%e 84: (4,2,1,1)
%e 96: (2,1,1,1,1,1)
%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[conj[primeMS[#]],_?OddQ]&]
%Y These partitions are counted by A045931.
%Y The conjugate strict version is counted by A239241.
%Y The conjugate version is A325698.
%Y These are the positions of 0's in A350941.
%Y Adding the conjugate condition gives A350946, all four equal A350947.
%Y A257991 counts odd parts, conjugate A344616.
%Y A257992 counts even parts, conjugate A350847.
%Y A325698: # of even parts = # of odd parts.
%Y A349157: # of even parts = # of odd conjugate parts, counted by A277579.
%Y A350848: # even conjugate parts = # odd conjugate parts, counted by A045931.
%Y A350943: # of even conjugate parts = # of odd parts, counted by A277579.
%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 A000041 = integer partitions, strict A000009.
%Y A056239 adds up prime indices, counted by A001222, row sums of A112798.
%Y A316524 = alternating sum of prime indices, reverse A344616.
%Y Cf. A024619, A026424, A028260, A103919, A130780, A171966, A195017, A241638, A325700, A350849, A350942, A350949.
%K nonn
%O 1,2
%A _Gus Wiseman_, Jan 27 2022
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