Sociable Chain/Examples/12,496

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Example of Sociable Chain

The sociable chain whose smallest element is $12 \, 496$ is of order $5$.

It goes:

$12 \, 496 \to 14 \, 288 \to 15 \, 472 \to 14 \, 536 \to 14 \, 264 \to 12 \, 496$


Proof

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

By definition:

$\map s n = \map \sigma n - n$

where $\map \sigma n$ denotes the $\sigma$ function.


Thus:

\(\displaystyle \map s {12 \, 496}\) \(=\) \(\displaystyle \map \sigma {12 \, 496} - 12 \, 496\)
\(\displaystyle \) \(=\) \(\displaystyle 26 \, 784 - 12 \, 496\) $\sigma$ of $12 \, 496$
\(\displaystyle \) \(=\) \(\displaystyle 14 \, 288\)


\(\displaystyle \map s {14 \, 288}\) \(=\) \(\displaystyle \map \sigma {14 \, 288} - 14 \, 288\)
\(\displaystyle \) \(=\) \(\displaystyle 29 \, 760 - 14 \, 288\) $\sigma$ of $14 \, 288$
\(\displaystyle \) \(=\) \(\displaystyle 15 \, 472\)


\(\displaystyle \map s {15 \, 472}\) \(=\) \(\displaystyle \map \sigma {15 \, 472} - 15 \, 472\)
\(\displaystyle \) \(=\) \(\displaystyle 30 \, 008 - 15 \, 472\) $\sigma$ of $15 \, 472$
\(\displaystyle \) \(=\) \(\displaystyle 14 \, 536\)


\(\displaystyle \map s {14 \, 536}\) \(=\) \(\displaystyle \map \sigma {14 \, 536} - 14 \, 536\)
\(\displaystyle \) \(=\) \(\displaystyle 28 \, 800 - 14 \, 536\) $\sigma$ of $14 \, 536$
\(\displaystyle \) \(=\) \(\displaystyle 14 \, 264\)


\(\displaystyle \map s {14 \, 264}\) \(=\) \(\displaystyle \map \sigma {14 \, 264} - 14 \, 264\)
\(\displaystyle \) \(=\) \(\displaystyle 26 \, 760 - 14 \, 264\) $\sigma$ of $14 \, 264$
\(\displaystyle \) \(=\) \(\displaystyle 12 \, 496\)


It is interesting to list the prime decomposition of each of the terms in the chain:

\(\displaystyle 12 \, 496\) \(=\) \(\displaystyle 2^4 \times 11 \times 71\)
\(\displaystyle 14 \, 288\) \(=\) \(\displaystyle 2^4 \times 19 \times 47\)
\(\displaystyle 15 \, 472\) \(=\) \(\displaystyle 2^4 \times 967\)
\(\displaystyle 14 \, 536\) \(=\) \(\displaystyle 2^3 \times 23 \times 79\)
\(\displaystyle 14 \, 264\) \(=\) \(\displaystyle 2^3 \times 1783\)

$\blacksquare$


Historical Note

The sociable chain of $12 \, 496$ was discovered by Paul Poulet.

He included a reference to it in a paper he published in $1918$.


Sources