In the pre-MO era, I once realized that on the integers, the function

$$ d(m, n) := \sqrt{\log \frac{\sqrt{mn}} {\text{gcd}(m,n)}}\ , $$

is a metric (all properties are easily verified; in fact this is a Hilbertian metric).

Now that I got reminded of it, I wanted to ask

Has anyone seen this metric before? If so, in what context? What could be potential applications of this metric?