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 $T(1053779)$.