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Inscribed circles of triangles with one angle of 60 degrees

Problem 195

Published on Friday, 23rd May 2008, 02:00 pm; Solved by 731

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 rn, so
T(100) = 1234,  T(1000) = 22767, and  T(10000) = 359912.

Find T(1053779).