Equivalence of Definitions of Quasiamicable Numbers

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

Proof
Let $\map s n$ 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 $\map {\sigma_1} n$, where $\sigma_1$ is the divisor sum function.

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

Thus:
 * $\map s n = \map {\sigma_1} n - n - 1$

Suppose:
 * $\map s n = m$

and:
 * $\map s m = n$

Then:

Similarly:

Thus:
 * $\map s n = \map s m = m + n + 1$

The argument reverses.