A350848 - OEIS

%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