$ \sigma_k(F_m) = F_n $

Let $\sigma_k(n)$ be the sum of the $k$th powers of the divisors of $n$. Here, we prove that if $(F_n)_{n \geq 1}$ is the Fibonacci sequence, then the only solutions of the equation $\sigma_k(F_m) = F_n$ in positive integers $k \geq 2$, $m$ and $n$ have $k=2$ and $m \in \{1,2,3\}$. The proof uses linear forms in two and three logarithms, lattice basis reduction, and some elementary considerations