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

> Does anyone know about applications where such a metric can be useful?

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**PS:** I remember that this metric had some relations to [Addition chains][1]. 


  [1]: http://en.wikipedia.org/wiki/Addition_chain