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A349157 Heinz numbers of integer partitions where the number of even parts is equal to the number of odd conjugate parts.

%I A349157 #7 Jan 27 2022 20:46:03
%S A349157 1,4,6,15,16,21,24,25,35,60,64,77,84,90,91,96,100,121,126,140,143,150,
%T A349157 210,221,240,247,256,289,297,308,323,336,351,360,364,375,384,400,437,
%U A349157 462,484,490,495,504,525,529,546,551,560,572,585,600,625,667,686,726
%N A349157 Heinz numbers of integer partitions where the number of even parts is equal to the number of odd conjugate parts.
%C A349157 The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are numbers with the same number of even prime indices as odd conjugate prime indices.
%C A349157 These are also partitions for which the number of even parts is equal to the positive alternating sum of the parts.
%F A349157 A257992(a(n)) = A257991(A122111(a(n))).
%e A349157 The terms and their prime indices begin:
%e A349157     1: ()
%e A349157     4: (1,1)
%e A349157     6: (2,1)
%e A349157    15: (3,2)
%e A349157    16: (1,1,1,1)
%e A349157    21: (4,2)
%e A349157    24: (2,1,1,1)
%e A349157    25: (3,3)
%e A349157    35: (4,3)
%e A349157    60: (3,2,1,1)
%e A349157    64: (1,1,1,1,1,1)
%e A349157    77: (5,4)
%e A349157    84: (4,2,1,1)
%e A349157    90: (3,2,2,1)
%e A349157    91: (6,4)
%e A349157    96: (2,1,1,1,1,1)
%t A349157 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A349157 conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
%t A349157 Select[Range[100],Count[primeMS[#],_?EvenQ]==Count[conj[primeMS[#]],_?OddQ]&]
%Y A349157 A subset of A028260 (even bigomega), counted by A027187.
%Y A349157 These partitions are counted by A277579.
%Y A349157 This is the half-conjugate version of A325698, counted by A045931.
%Y A349157 A000041 counts partitions, strict A000009.
%Y A349157 A047993 counts balanced partitions, ranked by A106529.
%Y A349157 A056239 adds up prime indices, row sums of A112798, counted by A001222.
%Y A349157 A100824 counts partitions with at most one odd part, ranked by A349150.
%Y A349157 A108950/A108949 count partitions with more odd/even parts.
%Y A349157 A122111 represents conjugation using Heinz numbers.
%Y A349157 A130780/A171966 count partitions with more or equal odd/even parts.
%Y A349157 A257991/A257992 count odd/even prime indices.
%Y A349157 A316524 gives the alternating sum of prime indices (reverse: A344616).
%Y A349157 Cf. A000700, A000712, A035363, A066207, A066208, A097613, A215366, A239241, A240009, A241638, A316523, A325700, A340604.
%K A349157 nonn
%O A349157 1,2
%A A349157 _Gus Wiseman_, Jan 21 2022