Equivalence of Definitions of Quasiamicable Numbers

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Theorem

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


The following definitions of the concept of Quasiamicable Numbers are equivalent:

Definition 1

$m$ and $n$ are quasiamicable numbers if and only if:

the sum of the proper divisors of $m$ is equal to $n$

and:

the sum of the proper divisors of $n$ is equal to $m$.

Definition 2

$m$ and $n$ are quasiamicable numbers if and only if:

$\map {\sigma_1} m = \map {\sigma_1} n = m + n + 1$

where $\sigma_1$ denotes the divisor sum function.


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:

\(\ds \map {\sigma_1} n - n - 1\) \(=\) \(\ds m\) Definition of Proper Divisor of Integer
\(\ds \leadsto \ \ \) \(\ds \map {\sigma_1} n\) \(=\) \(\ds m + n + 1\)


Similarly:

\(\ds \map {\sigma_1} m - m - 1\) \(=\) \(\ds n\) Definition of Proper Divisor of Integer
\(\ds \leadsto \ \ \) \(\ds \map {\sigma_1} m\) \(=\) \(\ds m + n + 1\)

Thus:

$\map s n = \map s m = m + n + 1$


The argument reverses.

$\blacksquare$