The starting point of this question is the observation that the smallest positive integers $a,b,c$ satisfying

$$\frac{a}{b+c} + \frac{b}{a+c} + \frac{c}{a+b} = 4$$

are [absurdly high][1]. This prompts the following general question: Is the set $$ C = \{n\in\mathbb{N}\setminus\{0\}: (\exists a,b,c \in\mathbb{N}\setminus\{0\}):\frac{a}{b+c} + \frac{b}{a+c} + \frac{c}{a+b} = n\}$$
computable? Is it even [recursive][2]?


  [1]: https://plus.google.com/+johncbaez999/posts/Pr8LgYYxvbM
  [2]: https://en.wikipedia.org/wiki/Recursive_set