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.
$\blacksquare$