Here, as my contribution, I add these conjectures (I don't know if these are in the literature) for a subset of the lattice $\mathbb{Z}_{\geq 1}\times\mathbb{Z}_{\geq 1}$. My computational evidence is small, I hope that it is interesting for you or for the professors and users of this site. Of course I don't know how solve the question.

As usual we denote the sum of positive divisors function of an integer $n\geq 1$ as $\sigma(n)=\sum_{1\leq d\mid n}d$, that is an important multiplicative function. I add the Wikipedia *Divisor function.*

**Conjecture 1.** *If* $x\geq 2$ *and* $y\geq 1$ *are both integers, and satisfy*
$$\sigma(\sigma(xy)-xy)=x,\tag{1}$$
*then* $y=1$.

**Remark.** In previous conjecture, and it applies also for the following conjectures, one can to express the consequence of the conjecture (I say from the computational evidence that I got) as $x-1$ *is a Mersenne prime* instead of the claimed consequence/conjecture $y=1$.

**Conjecture 2.** *If* $x\geq 2$ *and* $y\geq 1$ *are both integers, and these numbers satisfy*
$$\sigma(\sigma(xy)-(x+y-1))=x,\tag{2}$$
*then* $y=1$.

**Conjecture 3.** *Let* $x\geq 2$ *and* $y\geq 1$ *be integers such that* $\sigma(x+y-1)>xy$. *If the equation*
$$\sigma(\sigma(x+y-1)-xy)=x,\tag{3}$$
*holds, then* $y=1$.

I refer below the reference for the equation, and I add that one knows (if it can be inspiring here) also the article *Variations on Euclid's formula for perfect numbers*, by Farideh Firoozbakht and Maximilian F. Hasler, from Journal of Integer Sequences (2010) Volume: 13, Issue: 3, Article 10.3.1.

## References:

[1] Sequence *A072868*, from The On-Line Encyclopedia of Integer Sequences.

1more comment