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 $\map s n$ denote the aliquot sum of $n$.

By definition:

$\map s n = \map {\sigma_1} n - n$

where $\sigma_1$ denotes the divisor sum function.


Thus:

\(\ds \map s {103 \, 340 \, 640}\) \(=\) \(\ds \map {\sigma_1} {103 \, 340 \, 640} - 103 \, 340 \, 640\)
\(\ds \) \(=\) \(\ds 350 \, 584 \, 416 - 103 \, 340 \, 640\) $\sigma_1$ of $103 \, 340 \, 640$
\(\ds \) \(=\) \(\ds 247 \, 243 \, 776\)
\(\ds \) \(=\) \(\ds 123 \, 228 \, 768 + 124 \, 015 \, 008\)


\(\ds \map s {123 \, 228 \, 768}\) \(=\) \(\ds \map {\sigma_1} {123 \, 228 \, 768} - 123 \, 228 \, 768\)
\(\ds \) \(=\) \(\ds 350 \, 584 \, 416 - 123 \, 228 \, 768\) $\sigma_1$ of $123 \, 228 \, 768$
\(\ds \) \(=\) \(\ds 227 \, 355 \, 648\)
\(\ds \) \(=\) \(\ds 103 \, 340 \, 640 + 124 \, 015 \, 008\)


\(\ds \map s {124 \, 015 \, 008}\) \(=\) \(\ds \map {\sigma_1} {124 \, 015 \, 008} - 124 \, 015 \, 008\)
\(\ds \) \(=\) \(\ds 350 \, 584 \, 416 - 124 \, 015 \, 008\) $\sigma_1$ of $124 \, 015 \, 008$
\(\ds \) \(=\) \(\ds 226 \, 569 \, 408\)
\(\ds \) \(=\) \(\ds 103 \, 340 \, 640 + 123 \, 228 \, 768\)

$\blacksquare$


Sources

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