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A350943 Heinz numbers of integer partitions of which the number of even conjugate parts is equal to the number of odd parts.

%I A350943 #6 Jan 29 2022 12:49:40
%S A350943 1,3,6,7,13,14,18,19,26,27,29,36,37,38,42,43,53,54,58,61,63,70,71,74,
%T A350943 78,79,84,86,89,101,105,106,107,113,114,117,122,126,130,131,139,140,
%U A350943 142,151,156,158,162,163,171,173,174,178,181,190,193,195,199,202,210
%N A350943 Heinz numbers of integer partitions of which the number of even conjugate parts is equal to the number of odd parts.
%C A350943 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 A350943 A350847(a(n)) = A257991(a(n)).
%e A350943 The terms together with their prime indices begin:
%e A350943    1: ()
%e A350943    3: (2)
%e A350943    6: (2,1)
%e A350943    7: (4)
%e A350943   13: (6)
%e A350943   14: (4,1)
%e A350943   18: (2,2,1)
%e A350943   19: (8)
%e A350943   26: (6,1)
%e A350943   27: (2,2,2)
%e A350943   29: (10)
%e A350943   36: (2,2,1,1)
%e A350943   37: (12)
%e A350943   38: (8,1)
%e A350943   42: (4,2,1)
%e A350943 For example, the partition (6,3,2) has conjugate (3,3,2,1,1,1) and 1 = 1 so 195 is in the sequence.
%t A350943 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A350943 conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
%t A350943 Select[Range[100],Count[primeMS[#],_?OddQ]==Count[conj[primeMS[#]],_?EvenQ]&]
%Y A350943 These partitions are counted by A277579.
%Y A350943 The conjugate version is A349157, also counted by A277579.
%Y A350943 These are the positions of 0's in A350942.
%Y A350943 A000041 = integer partitions, strict A000009.
%Y A350943 A056239 adds up prime indices, counted by A001222, row sums of A112798.
%Y A350943 A122111 = conjugation using Heinz numbers.
%Y A350943 A257991 = # of odd parts, conjugate A344616.
%Y A350943 A257992 = # of even parts, conjugate A350847.
%Y A350943 A316524 = alternating sum of prime indices.
%Y A350943 The following rank partitions:
%Y A350943   A325040: product = product of conjugate, counted by A325039.
%Y A350943   A325698: # of even parts = # of odd parts, counted by A045931.
%Y A350943   A350848: # of even conj parts = # of odd conj parts, counted by A045931.
%Y A350943   A350944: # of odd parts = # of odd conjugate parts, counted by A277103.
%Y A350943   A350945: # of even parts = # of even conjugate parts, counted by A350948.
%Y A350943 Cf. A000070, A000290, A027187, A027193, A103919, A236559, A344607, A344651, A345196, A350950, A350951.
%K A350943 nonn
%O A350943 1,2
%A A350943 _Gus Wiseman_, Jan 28 2022