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%I #7 Jan 29 2022 12:49:34
%S 0,1,0,1,1,0,0,3,-2,1,1,2,0,0,-1,3,1,0,0,3,-2,1,1,2,-1,0,0,2,0,1,1,5,
%T -1,1,-2,0,0,0,-2,3,1,0,0,3,1,1,1,4,-4,1,-1,2,0,0,-1,2,-2,0,1,1,0,1,0,
%U 5,-2,1,1,3,-1,0,0,2,1,0,1,2,-3,0,0,5,-2,1
%N Number of odd parts minus number of even conjugate parts 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). This gives a bijective correspondence between positive integers and integer partitions.
%e First positions n such that a(n) = 6, 5, 4, 3, 2, 1, 0, -1, -2, -3, -4, -5, -6, together with their prime indices, are:
%e 192: (2,1,1,1,1,1,1)
%e 32: (1,1,1,1,1)
%e 48: (2,1,1,1,1)
%e 8: (1,1,1)
%e 12: (2,1,1)
%e 2: (1)
%e 1: ()
%e 15: (3,2)
%e 9: (2,2)
%e 77: (5,4)
%e 49: (4,4)
%e 221: (7,6)
%e 169: (6,6)
%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[primeMS[n],_?OddQ]-Count[conj[primeMS[n]],_?EvenQ],{n,100}]
%Y The conjugate version is A350849.
%Y This is a hybrid of A195017 and A350941.
%Y Positions of 0's are A350943.
%Y A000041 = integer partitions, strict A000009.
%Y A056239 adds up prime indices, counted by A001222, row sums of A112798.
%Y A122111 represents conjugation using Heinz numbers.
%Y A257991 = # of odd parts, conjugate A344616.
%Y A257992 = # of even parts, conjugate A350847.
%Y A316524 = alternating sum of prime indices.
%Y The following rank partitions:
%Y A325698: # of even parts = # of odd parts.
%Y A349157: # of even parts = # of odd conjugate parts, counted by A277579.
%Y A350848: # even conj parts = # 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 Cf. A026424, A028260, A130780, A171966, A239241, A241638, A325700, A350947, A350949, A350950, A350951.
%K sign
%O 1,8
%A _Gus Wiseman_, Jan 28 2022
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