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 $$ C = \{n\in\mathbb{N}\setminus\{0\}: (\exists a,b,c \in\mathbb{N}\setminus\{0\}):\frac{a}{b+c} + \frac{b}{a+c} + \frac{c}{a+b} = n\}$$ computable? (As user Watson points out in the comment section below, $C$ contains no odd numbers. It would also be great to see an even number $\geq 6$ not contained in $C$.)