# Computability of a “weird” set

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. This leads to the following general question: Is the set $C\subseteq {\mathbb N}$ defined by $$C = \left\{n\in\mathbb{N}\setminus\{0\}: \big(\exists a,b,c \in\mathbb{N}\setminus\{0\}\big):\frac{a}{b+c} + \frac{b}{a+c} + \frac{c}{a+b} = n\right\}$$ computable?