Problem 397

Triangle on parabola

On the parabola `y` = `x`^{2}/`k`, three points A(`a`, `a`^{2}/`k`), B(`b`, `b`^{2}/`k`) and C(`c`, `c`^{2}/`k`) are chosen.

Let F(`K`, `X`) be the number of the integer quadruplets (`k`, `a`, `b`, `c`) such that at least one angle of the triangle ABC is 45-degree, with 1 `k` `K` and -`X` `a` `b` `c` `X`.

For example, F(1, 10) = 41 and F(10, 100) = 12492.

Find F(10^{6}, 10^{9}).

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