# Definition:Amicable Triplet

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## Definition

Let $m_1, m_2, m_3 \in \Z_{>0}$ be (strictly) positive integers.

### Definition 1

$\tuple {m_1, m_2, m_3}$ are an amicable triplet if and only if the aliquot sum of any one of them equals the sum of the other two:

the aliquot sum of $m_1$ is equal to $m_2 + m_3$

and:

the aliquot sum of $m_2$ is equal to $m_1 + m_3$

and:

the aliquot sum of $m_3$ is equal to $m_1 + m_2$

### Definition 2

$\left({m_1, m_2, m_3}\right)$ are an amicable triplet if and only if:

$\sigma \left({m_1}\right) = \sigma \left({m_2}\right) = \sigma \left({m_3}\right) = m_1 + m_2 + m_3$

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

## Examples

### $1980$, $2016$ and $2556$

$\tuple {1980, 2016, 2556}$ form an amicable triplet.

### $103 \, 340 \, 640$, $123 \, 228 \, 768$ and $124 \, 015 \, 008$

The following numbers form an amicable triplet:

$103 \, 340 \, 640$
$123 \, 228 \, 768$
$124 \, 015 \, 008$

## Also known as

An amicable triplet is also known as an amicable triple.