Define a *pro-perfect set* $S$ to be a finite set of positive integers satisfying the following three properties:


1. $1\in S$.
2. $\displaystyle\sum\_{n\in S}n^{-1}\in S$
3. There exists a unique permutation $\pi$ of $S$ of order $2$ such that $$\frac{1}{\vert S\vert}\sum\_{n\in S}n\pi(n)\in S.$$


**Question 1**: Is any pro-perfect set necessarily the set of divisors of a perfect number? 


**Question 2**: Does every pro-perfect set necessarily contain $2$?


*Edit 1*: it might help to consider the more general notion of *$k$-pro-perfect set* such that property 3 is replaced by 


3' : There exists a (not necessarily unique) permutation $\pi$ of order $k$ such that $$\frac{1}{\vert S\vert}\sum\_{n\in S}\prod\_{\ell=0}^{k-1}\pi^{\circ \ell}(n)\in S.$$


*Edit 2*: Let $s\_{\text{inv}}$ denote the quantity $$s\_{\text{inv}}:=\frac{1}{\vert S\vert}\sum\_{n\in S}n\pi(n),$$ one gets the following equivalence: $$\forall n\in S, \quad n\mid s\_{inv}\;\Longleftrightarrow\;\pi:n\mapsto\frac{s\_{\text{inv}}}{n}.$$


Edit 3: let $s\_{rec}$ denote $\sum\_{s\in S}n^{-1}$ and suppose the $m$ first elements of $S$ when arranged in increasing order form a geometric progression with constant ratio $q$. Then $s\_{rec}\geq\frac{1-q^{-m}}{1-q^{-1}}$. The latter is less than $2$ as $q\geq 2$. Moreover $s\_{rec}\leq\frac{\vert S\vert(1-q^{-m)}}{m(1-q^{-1})}<\frac{2\vert S\vert}{m}$.



A possible axiomatic characterization of the set of divisors of a perfect number
Try this.


$$
S = \{1,2,3,12,18,36\}
$$
then
$$
\sum\_{k \in S}\frac{1}{k} = 2 \in S
$$
and
$$
\frac{1}{6}\sum\_{k \in S} k \pi(k) = 36 \in S
$$
when $\pi$ is the permutation of $S$ that reverses the order, while
$$
\frac{1}{6}\sum\_{k \in S} k \pi(k) > 36
$$
for any other permutation $\pi$.


Uniqueness of the minimizing permutation is from [Hardy-Littlewood-Polya rearrangement](https://en.wikipedia.org/wiki/Rearrangement_inequality).


