Equivalence of Definitions of Quasiamicable Numbers

Theorem
Let $m \in \Z_{>0}$ and $n \in \Z_{>0}$ be (strictly) positive integers.

Proof
Let $s \left({n}\right)$ denote the sum of the proper divisors of (strictly) positive integer $n$.

The sum of all the divisors of a (strictly) positive integer $n$ is $\sigma \left({n}\right)$, where $\sigma$ is the $\sigma$ function.

The proper divisors of $n$ are the divisors $n$ with $1$ and $n$ excluded.

Thus:
 * $s \left({n}\right) = \sigma \left({n}\right) - n - 1$

Suppose:
 * $s \left({n}\right) = m$

and:
 * $s \left({m}\right) = n$

Then:

Similarly:

Thus:
 * $s \left({n}\right) = s \left({m}\right) = m + n + 1$

The argument reverses.