Definition:Amicable Triplet/Definition 2
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Definition
Let $m_1, m_2, m_3 \in \Z_{>0}$ be (strictly) positive integers.
$\tuple {m_1, m_2, m_3}$ 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 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