What is the density of Descartes numbers?

Lagrange showed that every natural number can be represented as the sum of 4 squares (of natural numbers). Let us call a natural number a Descartes number if it can be represented as the sum of the squares of 4 natural numbers, with the natural numbers being the curvatures of 4 circles in Descartes configuration (that is, that fit the hypothesis of the Descartes Circle Theorem).

That is, call a natural number $n$ a Descartes number if there exist natural numbers $k_1,k_2,k_3$ and $k_4$ such that $$n = k_1^2+k_2^2+k_3^2+k_4^2=\frac{1}{2}(k_1+k_2+k_3+k_4)^2.$$

Do Descartes numbers exist? More generally, what is their density?

• The first thing I'd want to have, to attack your question, is a parametric formula for all integer quadruples $(k_1,k_2,k_3,k_4)$ that satisfy that equation (similar to a parametric formula for all Pythagorean triples of integers). – Greg Martin Dec 2 '13 at 22:55