Equivalence of Definitions of Amicable Triplet

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

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

Then:

Similarly:

and:

Thus:
 * $\map {\sigma_1} {m_1} = \map {\sigma_1} {m_2} = \map {\sigma_1} {m_3} = m_1 + m_2 + m_3$

The argument reverses.