Is it always true that if $\sigma(n) + 1$ is divisible by $n$ then $\sigma(n) = 2n - 1$?

Question

Suppose $n$ is a natural number, such that $\sigma(n) + 1$ is divisible by $n$. Is it always true that $\sigma(n) = 2n - 1$?

I checked this for all numbers less than $1000000$ and did not find any counterexample among them. However, any proof of this statement is also unknown to me.

Any help will be appreciated.