Calculating $n$ for $\sigma(\sigma(n)-n) = n$ [redefined] As in A072868 described by OEIS;
Defined by $\sigma(\sigma(n)-n) = n$.
Since these numbers are important in regard to many things, specially mersenne primes, since  ${n-1 \over 2}\times n,~\sigma(\sigma(n)-n) = n$.
Expression explained: $n$ minus $1$ divided by $2$ is a perfect number whenever $\sigma(\sigma(n)-n) = n$
Is there any known way to calculate $n$ for $\sigma(\sigma(n)-n) = n$, or is this yet to be discovered?
 A: 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.
