Definition:Amicable Triplet

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

$\tuple {m_1, m_2, m_3}$ are an amicable triplet if and only if:

$\map {\sigma_1} {m_1} = \map {\sigma_1} {m_2} = \map {\sigma_1} {m_3} = m_1 + m_2 + m_3$

where $\sigma_1$ denotes the divisor sum 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.

Also see

• Equivalence of Definitions of Amicable Triplet

• Results about amicable triplets can be found here.

Sources

• Weisstein, Eric W. "Amicable Triple." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/AmicableTriple.html