Obviously, the most interesting pythagorean triple $(a, b, c)$ would be one for which the corresponding triangle (with integer side lengths $a, b, c$) has angles 90°, 60° and 30° ($\frac{\pi}{2}, \frac{\pi}{3}$ and $\frac{\pi}{6}$). This would mean that $c = 2a$ (since $\sin \frac{\pi}{6} = \frac{1}{2}$).

But in case this doens't exist, I would be interested to learn about any triple leading to "nice" or "interesting" acute angles.