13.444 The values of the lengths of the sides of a triangle and its area are four consecutive integers, respectively. Find the lengths of the sides of this triangle.
Solution. According to the condition, the sides of the triangle are $n-1, n, n+1$, and the area is $n+2$, where $n$ is a natural number. By Heron's formula,
$n+2=\sqrt{\frac{3 n}{2} \cdot\left(\frac{3 n}{2}-n\right) \cdot\left(\frac{3 n}{2}-n+1\right) \cdot\left(\frac{3 n}{2}-n-1\right)} \Rightarrow 16(n+2)=3 n^{2}(n-2)$

From this, $n$ is even, i.e., $n=2 k$. Therefore, $4(k+1)=3 k^{2}(k-1)$.

Since $k$ and $k+1$ are coprime, 4 is divisible by $k^{2} \Rightarrow k^{2}=4 \Rightarrow n=4$.

Answer: $3,4,5$.