Amicable Triplet/Examples/103,340,640-123,228,768-124,015,008

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Example of Amicable Triplet

The following numbers form an amicable triplet:

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


Proof

Let $s \left({n}\right)$ denote the aliquot sum of $n$.

By definition:

$s \left({n}\right) = \sigma \left({n}\right) - n$

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


Thus:

\(\displaystyle s \left({103 \, 340 \, 640}\right)\) \(=\) \(\displaystyle \sigma \left({103 \, 340 \, 640}\right) - 103 \, 340 \, 640\)
\(\displaystyle \) \(=\) \(\displaystyle 350 \, 584 \, 416 - 103 \, 340 \, 640\) $\sigma$ of $103 \, 340 \, 640$
\(\displaystyle \) \(=\) \(\displaystyle 247 \, 243 \, 776\)
\(\displaystyle \) \(=\) \(\displaystyle 123 \, 228 \, 768 + 124 \, 015 \, 008\)


\(\displaystyle s \left({123 \, 228 \, 768}\right)\) \(=\) \(\displaystyle \sigma \left({123 \, 228 \, 768}\right) - 123 \, 228 \, 768\)
\(\displaystyle \) \(=\) \(\displaystyle 350 \, 584 \, 416 - 123 \, 228 \, 768\) $\sigma$ of $123 \, 228 \, 768$
\(\displaystyle \) \(=\) \(\displaystyle 227 \, 355 \, 648\)
\(\displaystyle \) \(=\) \(\displaystyle 103 \, 340 \, 640 + 124 \, 015 \, 008\)


\(\displaystyle s \left({124 \, 015 \, 008}\right)\) \(=\) \(\displaystyle \sigma \left({124 \, 015 \, 008}\right) - 124 \, 015 \, 008\)
\(\displaystyle \) \(=\) \(\displaystyle 350 \, 584 \, 416 - 124 \, 015 \, 008\) $\sigma$ of $124 \, 015 \, 008$
\(\displaystyle \) \(=\) \(\displaystyle 226 \, 569 \, 408\)
\(\displaystyle \) \(=\) \(\displaystyle 103 \, 340 \, 640 + 123 \, 228 \, 768\)

$\blacksquare$


Sources