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A350849 Number of odd conjugate parts minus number of even parts in the integer partition with Heinz number n.

%I A350849 #10 Jan 29 2022 12:49:17
%S A350849 0,1,1,0,3,0,3,1,-2,2,5,1,5,2,0,0,7,-1,7,3,0,4,9,0,0,4,-1,3,9,1,11,1,
%T A350849 2,6,0,-2,11,6,2,2,13,1,13,5,1,8,15,1,-2,1,4,5,15,-2,2,2,4,8,17,0,17,
%U A350849 10,1,0,2,3,19,7,6,1,19,-1,21,10,1,7,0,3,21,3
%N A350849 Number of odd conjugate parts minus number of even parts in the integer partition with Heinz number n.
%C A350849 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 A350849 a(n) = A344616(n) - A257992(n).
%e A350849 First positions n such that a(n) = 4, 3, 2, 1, 0, -1, -2, -3, -4, together with their prime indices, are:
%e A350849    22: (5,1)
%e A350849     5: (3)
%e A350849    10: (3,1)
%e A350849     2: (1)
%e A350849     1: ()
%e A350849    18: (2,2,1)
%e A350849     9: (2,2)
%e A350849   162: (2,2,2,2,1)
%e A350849    81: (2,2,2,2)
%t A350849 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A350849 conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
%t A350849 Table[Count[conj[primeMS[n]],_?OddQ]-Count[primeMS[n],_?EvenQ],{n,100}]
%Y A350849 This is a hybrid of A195017 and A350941.
%Y A350849 Positions of 0's are A349157.
%Y A350849 Counting even conjugate parts instead of even parts gives A350941.
%Y A350849 The conjugate version is A350942.
%Y A350849 A257991 counts odd parts, conjugate A344616.
%Y A350849 A257992 counts even parts, conjugate A350847.
%Y A350849 The following rank partitions:
%Y A350849   A325698: # of even parts = # of odd parts.
%Y A350849   A349157: # of even parts = # of odd conjugate parts, counted by A277579.
%Y A350849   A350848: # even conj parts = # odd conj parts, counted by A045931.
%Y A350849   A350943: # of even conjugate parts = # of odd parts, counted by A277579.
%Y A350849   A350944: # of odd parts = # of odd conjugate parts, counted by A277103.
%Y A350849   A350945: # of even parts = # of even conjugate parts, counted by A350948.
%Y A350849 A000041 = integer partitions, strict A000009.
%Y A350849 A056239 adds up prime indices, counted by A001222, row sums of A112798.
%Y A350849 A122111 represents conjugation using Heinz numbers.
%Y A350849 A316524 = alternating sum of prime indices.
%Y A350849 Cf. A026424, A028260, A130780, A171966, A239241, A241638, A325700, A350947, A350949, A350950, A350951.
%K A350849 sign
%O A350849 1,5
%A A350849 _Gus Wiseman_, Jan 28 2022