Definition:Quasiamicable Numbers
Definition
Let $m \in \Z_{>0}$ and $n \in \Z_{>0}$ be (strictly) positive integers.
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.
Sequence
The sequence of quasiamicable pairs begins:
- $\tuple {48, 75}, \tuple {140, 195}, \tuple {1050, 1925}, \tuple {1575, 1648} \ldots$
Examples
$48$ and $75$
$48$ and $75$ form a quasiamicable pair.
$140$ and $195$
$140$ and $195$ form a quasiamicable pair.
$1050$ and $1925$
$1050$ and $1925$ form a quasiamicable pair.
$1575$ and $1648$
$1575$ and $1648$ form a quasiamicable pair.
Also known as
Quasiamicable numbers are often referred to as quasiamicable pairs, which acknowledges the fact that they come in sets of $2$ at a time.
Some sources hyphenate: quasi-amicable.
They are often seen with the name betrothed numbers.
Some sources refer to quasiamicable pairs as reduced amicable pairs.
Also see
Sources
- Weisstein, Eric W. "Quasiamicable Pair." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/QuasiamicablePair.html