Let $r_1,r_2\dots$ be the radii of [Apollonian gasket][1].
I would like to know for which values $\alpha$ we have
$$\sum_{n=1}^\infty r_n^\alpha<\infty.$$

![Integral Apollonian circle packing defined by circle curvatures of (−1, 2, 2, 3)][2]

I know that if three circles $A$, $B$ and $C$ are tangent to two circles $D_1$ and $D_2$ then 
$$d_1+d_2=2(a+b+c),$$
where $a$, $b$, $c$, $d_1$ and $d_2$ denote the curvatures of the corresponding circle.
(For example, on the picture, $3+3=2(2+2-1)$.)

In principle, it gives a recursive formula for $r_n$, but I was not able to figure out how to use it.

**Motivation:** I would like to know if one can cover whole measure of a square with countable number of disjoint open discs with radii  $r_1,r_2\dots$ such that 
$$\sum_{n=1}^\infty r_n^\alpha<\infty$$ 
for all $\alpha>1$. 
By some reason I believe that Apollonian gasket is optimal in this sense; at least it worth to check it.


  [1]: http://en.wikipedia.org/wiki/Apollonian_gasket
  [2]: https://upload.wikimedia.org/wikipedia/commons/thumb/7/7a/ApollonianGasket-1_2_2_3-Labels.png/480px-ApollonianGasket-1_2_2_3-Labels.png