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f[n_] := Module[{d, count, x, s, ys}, d = Divisors[n]; count = 0; Do[If[ IntegerQ[Sqrt[x^4 + 4n x]], s = Sqrt[x^4 + 4n x]; ys = Select[{-(s+x^2)/ (2x), (x^2-s)/(2x)}, IntegerQ[#] && GCD[#, x] == 1&]; count = count + Length[ys]], {x, Union[d, -d]}]; count]; Array[f, 200] (* Jean-François Alcover, Apr 29 2019, after Robert Israel *)
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Robert Israel, <a href="/A297968/b297968_1.txt">Table of n, a(n) for n = 1..10000</a>
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for x in d union map(`-`, d) do
0, 2, 4, 0, 0, 0, 4, 6, 0, 0, 0, 0, 0, 4, 6, 0, 0, 0, 0, 0, 0, 0, 4, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 6, 0, 4, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 6, 0, 0, 0, 0, 0, 4, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0
Robert Israel, <a href="/A297968/b297968_1.txt">Table of n, a(n) for n = 1..10000</a>
For n=6 the a(n)=6 solutions are (x,y) = (-3,1), (-3,2), (1,-3), (1,2), (2,1) and (2,-3).
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a(n)=0 if n is odd. - Robert Israel, Jan 10 2018
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