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%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
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