Equivalence of Definitions of Amicable Pair

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

Definition 1 is equivalent to Definition 2
Let $\map s n$ denote the aliquot sum 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 aliquot sum of $n$ is the sum of the divisors of $n$ with $n$ excluded.

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

Suppose:
 * $\map s n = m$

and:
 * $\map s m = n$

Then:

Similarly:

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

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:
 * $\map s {a_0} = a_1$
 * $\map s {a_1} = a_0$

and the equivalence follows.