# Definition:Quasiamicable Numbers

## Contents

## 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

- Weisstein, Eric W. "Quasiamicable Pair." From
*MathWorld*--A Wolfram Web Resource. http://mathworld.wolfram.com/QuasiamicablePair.html