Let $\sigma(n)$ denote the sum of the divisors of $n$. (https://oeis.org/A000203)
It is relatively easy to find numbers $n$ such that $f(g(n)) = g(f(n))$ where $f(n) = \sigma(n)$ and $g(n) = \sigma(n) -n$ (https://oeis.org/A291881) when $n$ is even.
Question. What is the smallest odd number $n \ge 1$ such that $\sigma(\sigma(n)) = \sigma(\sigma(n)-n)+\sigma(n)$ (if such an odd number exists) ?
If someone can show that there is no such odd $n$ or there is any reference to this spesific equation, that answer is also very welcome.