Equivalence of Definitions of Amicable Triplet

Theorem
Let $m_1, m_2, m_3 \in \Z_{>0}$ be (strictly) positive integers.

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

Proof
For $n \in \Z_{>0}$, 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:

Then:

Similarly:

and:

Thus:
 * $\sigma \left({m_1}\right) = \sigma \left({m_2}\right) = \sigma \left({m_3}\right) = m_1 + m_2 + m_3$

The argument reverses.