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