Equivalence of Definitions of Amicable Pair

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

The following definitions for $m$ and $n$ to be amicable numbers are equivalent:

Definition 1 is equivalent to Definition 2
Let $s \left({n}\right)$ denote the aliquot sum 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 aliquot sum of $n$ is the sum of the divisors of $n$ with $n$ excluded.

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

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

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

Then:

Similarly:

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

The argument reverses.

Definition 1 is equivalent to Definition 3
From the definition of definition 1 of an amicable pair:

From the definition of a sociable chain:

Here it is seen that setting $r = 2$ gives that:
 * $s \left({a_0}\right) = a_1$
 * $s \left({a_1}\right) = a_0$

and the equivalence follows.