Definition:Quasiamicable Numbers

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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:

$\sigma \left({m}\right) = \sigma \left({n}\right) = m + n + 1$

where $\sigma \left({m}\right)$ denotes the $\sigma$ 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