A349157 - OEIS

%I #7 Jan 27 2022 20:46:03

%S 1,4,6,15,16,21,24,25,35,60,64,77,84,90,91,96,100,121,126,140,143,150,

%T 210,221,240,247,256,289,297,308,323,336,351,360,364,375,384,400,437,

%U 462,484,490,495,504,525,529,546,551,560,572,585,600,625,667,686,726

%N Heinz numbers of integer partitions where the number of even 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), so these are numbers with the same number of even prime indices as odd conjugate prime indices.

%C These are also partitions for which the number of even parts is equal to the positive alternating sum of the parts.

%F A257992(a(n)) = A257991(A122111(a(n))).

%e The terms and their prime indices begin:

%e     1: ()

%e     4: (1,1)

%e     6: (2,1)

%e    15: (3,2)

%e    16: (1,1,1,1)

%e    21: (4,2)

%e    24: (2,1,1,1)

%e    25: (3,3)

%e    35: (4,3)

%e    60: (3,2,1,1)

%e    64: (1,1,1,1,1,1)

%e    77: (5,4)

%e    84: (4,2,1,1)

%e    90: (3,2,2,1)

%e    91: (6,4)

%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[primeMS[#],_?EvenQ]==Count[conj[primeMS[#]],_?OddQ]&]

%Y A subset of A028260 (even bigomega), counted by A027187.

%Y These partitions are counted by A277579.

%Y This is the half-conjugate version of A325698, counted by A045931.

%Y A000041 counts partitions, strict A000009.

%Y A047993 counts balanced partitions, ranked by A106529.

%Y A056239 adds up prime indices, row sums of A112798, counted by A001222.

%Y A100824 counts partitions with at most one odd part, ranked by A349150.

%Y A108950/A108949 count partitions with more odd/even parts.

%Y A122111 represents conjugation using Heinz numbers.

%Y A130780/A171966 count partitions with more or equal odd/even parts.

%Y A257991/A257992 count odd/even prime indices.

%Y A316524 gives the alternating sum of prime indices (reverse: A344616).

%Y Cf. A000700, A000712, A035363, A066207, A066208, A097613, A215366, A239241, A240009, A241638, A316523, A325700, A340604.

%K nonn

%O 1,2

%A _Gus Wiseman_, Jan 21 2022