Eccoci.

Let's call an integer sided triangle with exactly one angle of $60$ degrees a $60$-degree triangle.
Let $r$ be the radius of the inscribed circle of such a $60$-degree triangle.

There are $1234$ $60$-degree triangles for which $r \le 100$.
Let $T(n)$ be the number of $60$-degree triangles for which $r \le n$, so
$T(100) = 1234$, $T(1000) = 22767$, and $T(10000) = 359912$.

Find $T(1053779)$.