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A350941 Number of odd conjugate parts minus number of even conjugate parts in the integer partition with Heinz number n.

%I A350941 #7 Jan 29 2022 12:49:26
%S A350941 0,1,2,-1,3,0,4,1,-2,1,5,2,6,2,-1,-1,7,0,8,3,0,3,9,0,-3,4,2,4,10,1,11,
%T A350941 1,1,5,-2,-2,12,6,2,1,13,2,14,5,3,7,15,2,-4,-1,3,6,16,0,-1,2,4,8,17,
%U A350941 -1,18,9,4,-1,0,3,19,7,5,0,20,0,21,10,1,8,-3,4
%N A350941 Number of odd conjugate parts minus number of even conjugate parts in the integer partition with Heinz number n.
%C A350941 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 A350941 a(n) = A344616(n) - A350847(n).
%e A350941 First positions n such that a(n) = 4, 3, 2, 1, 0, -1, -2, -3, -4, together with their prime indices, are:
%e A350941    7: (4)
%e A350941    5: (3)
%e A350941    3: (2)
%e A350941    2: (1)
%e A350941    1: ()
%e A350941    4: (1,1)
%e A350941    9: (2,2)
%e A350941   25: (3,3)
%e A350941   49: (4,4)
%t A350941 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A350941 conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
%t A350941 Table[Count[conj[primeMS[n]],_?OddQ]-Count[conj[primeMS[n]],_?EvenQ],{n,1,50}]
%Y A350941 A hybrid with A195017 (non-conjugate version) is A350849, conjugate A350942.
%Y A350941 Positions of 0's are A350848, counted by A045931.
%Y A350941 A000041 = integer partitions, strict A000009.
%Y A350941 A056239 adds up prime indices, counted by A001222, row sums of A112798.
%Y A350941 A122111 represents conjugation using Heinz numbers.
%Y A350941 A257991 counts odd parts, conjugate A344616.
%Y A350941 A257992 counts even parts, conjugate A350847.
%Y A350941 A316524 = alternating sum of prime indices.
%Y A350941 The following rank partitions:
%Y A350941   A325698: # of even parts = # of odd parts.
%Y A350941   A349157: # of even parts = # of odd conjugate parts, counted by A277579.
%Y A350941   A350943: # of even conjugate parts = # of odd parts, counted by A277579.
%Y A350941   A350944: # of odd parts = # of odd conjugate parts, counted by A277103.
%Y A350941   A350945: # of even parts = # of even conjugate parts, counted by A350948.
%Y A350941 Cf. A026424, A028260, A130780, A171966, A239241, A241638, A325700, A350841, A350947, A350949, A350951.
%K A350941 sign
%O A350941 0,3
%A A350941 _Gus Wiseman_, Jan 28 2022