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A350847 Number of even parts in the conjugate of the integer partition with Heinz number n.

%I A350847 #5 Mar 16 2022 03:17:37
%S A350847 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 A350847 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 A350847 2,2,0,1,0,1,1,0,4,1,0,0,2,1,0,2,3,1,2
%N A350847 Number of even parts in the conjugate of the integer partition with Heinz number n.
%C A350847 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 A350847 a(n) = A344616(n) - A350941(n).
%F A350847 a(n) = A257992(A122111(n)).
%t A350847 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A350847 conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
%t A350847 Table[Count[conj[primeMS[n]],_?EvenQ],{n,100}]
%Y A350847 Positions of first appearances are A001248.
%Y A350847 The triangular version is A116482.
%Y A350847 Positions of zeros are A346635.
%Y A350847 Subtracting from the number of odd conjugate parts gives A350941.
%Y A350847 Subtracting from the number of odd parts gives A350942.
%Y A350847 Subtracting from the number of even parts gives A350950.
%Y A350847 There are four statistics:
%Y A350847 - A257991 = # of odd parts, conjugate A344616.
%Y A350847 - A257992 = # of even parts, conjugate A350847 (this sequence).
%Y A350847 There are six possible pairings of statistics:
%Y A350847 - A325698: # of even parts = # of odd parts, counted by A045931.
%Y A350847 - A349157: # of even parts = # of odd conjugate parts, counted by A277579.
%Y A350847 - A350848: # of even conj parts = # of odd conj parts, counted by A045931.
%Y A350847 - A350943: # of even conjugate parts = # of odd parts, counted by A277579.
%Y A350847 - A350944: # of odd parts = # of odd conjugate parts, counted by A277103.
%Y A350847 - A350945: # of even parts = # of even conjugate parts, counted by A350948.
%Y A350847 There are three possible double-pairings of statistics:
%Y A350847 - A350946, counted by A351977.
%Y A350847 - A350949, counted by A351976.
%Y A350847 - A351980, counted by A351981.
%Y A350847 The case of all four statistics equal is A350947, counted by A351978.
%Y A350847 A056239 adds up prime indices, counted by A001222, row sums of A112798.
%Y A350847 A122111 represents partition conjugation using Heinz numbers.
%Y A350847 Cf. A028260, A130780, A171966, A195017, A236559, A239241, A241638, A316524, A325700, A350849, A350951.
%K A350847 nonn
%O A350847 1,9
%A A350847 _Gus Wiseman_, Mar 14 2022