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Number of even parts in the conjugate of the integer partition with Heinz number n.
23

%I #5 Mar 16 2022 03:17:37

%S 0,0,0,1,0,1,0,0,2,1,0,0,0,1,2,1,0,1,0,0,2,1,0,1,3,1,0,0,0,1,0,0,2,1,

%T 3,2,0,1,2,1,0,1,0,0,0,1,0,0,4,2,2,0,0,1,3,1,2,1,0,2,0,1,0,1,3,1,0,0,

%U 2,2,0,1,0,1,1,0,4,1,0,0,2,1,0,2,3,1,2

%N Number of even parts in the conjugate of the integer partition with Heinz number n.

%C The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so a(n) counts even prime indices of n.

%F a(n) = A344616(n) - A350941(n).

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

%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 Table[Count[conj[primeMS[n]],_?EvenQ],{n,100}]

%Y Positions of first appearances are A001248.

%Y The triangular version is A116482.

%Y Positions of zeros are A346635.

%Y Subtracting from the number of odd conjugate parts gives A350941.

%Y Subtracting from the number of odd parts gives A350942.

%Y Subtracting from the number of even parts gives A350950.

%Y There are four statistics:

%Y - A257991 = # of odd parts, conjugate A344616.

%Y - A257992 = # of even parts, conjugate A350847 (this sequence).

%Y There are six possible pairings of statistics:

%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 - 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 There are three possible double-pairings of statistics:

%Y - A350946, counted by A351977.

%Y - A350949, counted by A351976.

%Y - A351980, counted by A351981.

%Y The case of all four statistics equal is A350947, counted by A351978.

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

%Y A122111 represents partition conjugation using Heinz numbers.

%Y Cf. A028260, A130780, A171966, A195017, A236559, A239241, A241638, A316524, A325700, A350849, A350951.

%K nonn

%O 1,9

%A _Gus Wiseman_, Mar 14 2022