In this post we denote the Dedekind psi function as $\psi(m)$ for integers $m\geq 1$. This is an important arithmetic fuction in several subjects of mathematics. As reference I add the Wikipedia Dedekind psi function, and [1]. On the other hand I add the reference that Wikipedia has the article Mersenne prime, and that I was inspired in the formula that defines the sequence A072868 from the On-Line Encyclopedia of Integer Sequences. The Dedekind psi function can be represented for a positive integer $m>1$ as $$\psi(m)=m\prod_{\substack{p\mid m\\p\text{ prime}}}\left(1+\frac{1}{p}\right)$$ with the definition $\psi(1)=1$. Claim. If we take $n=2^p$ with $2^p-1$ a Mersenne prime, then the equation $$\psi(2(\psi(n)-n)-1)=n\tag{1}$$ holds. Sketch of proof. Just direct computation using the mentioned representation for the Dedekind psi function.$\square$ I don't know if previous equation is in the literature, one can to state a similar equation than $(1)$ involving the sum of divisors function instead of the Dedekind psi function. Question. I would like to know if it is possible to prove of refute that if an integer $n\geq 2$ satisfies $(1)$ then $n-1$ is a Mersenne prime. Many thanks. With a Pari/GP script and for small segments of integers I have not found counterexamples. I'm asking what work can be done for previous question proving the conjecture, or if you can to find a counterexample, before I'm accepting an available answer. References: [1] Tom M. Apostol, Introduction to analytic number theory, Undergraduate Texts in Mathematics, New York-Heidelberg: Springer-Verlag (1976). Login To add answer/comment