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:
- $\map s n = \map {\sigma_1} n - n$
where $\sigma_1$ denotes the divisor sum function.
Thus:
\(\ds \map s {12 \, 496}\) | \(=\) | \(\ds \map {\sigma_1} {12 \, 496} - 12 \, 496\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 26 \, 784 - 12 \, 496\) | $\sigma_1$ of $12 \, 496$ | |||||||||||
\(\ds \) | \(=\) | \(\ds 14 \, 288\) |
\(\ds \map s {14 \, 288}\) | \(=\) | \(\ds \map {\sigma_1} {14 \, 288} - 14 \, 288\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 29 \, 760 - 14 \, 288\) | $\sigma_1$ of $14 \, 288$ | |||||||||||
\(\ds \) | \(=\) | \(\ds 15 \, 472\) |
\(\ds \map s {15 \, 472}\) | \(=\) | \(\ds \map {\sigma_1} {15 \, 472} - 15 \, 472\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 30 \, 008 - 15 \, 472\) | $\sigma_1$ of $15 \, 472$ | |||||||||||
\(\ds \) | \(=\) | \(\ds 14 \, 536\) |
\(\ds \map s {14 \, 536}\) | \(=\) | \(\ds \map {\sigma_1} {14 \, 536} - 14 \, 536\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 28 \, 800 - 14 \, 536\) | $\sigma_1$ of $14 \, 536$ | |||||||||||
\(\ds \) | \(=\) | \(\ds 14 \, 264\) |
\(\ds \map s {14 \, 264}\) | \(=\) | \(\ds \map {\sigma_1} {14 \, 264} - 14 \, 264\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 26 \, 760 - 14 \, 264\) | $\sigma_1$ of $14 \, 264$ | |||||||||||
\(\ds \) | \(=\) | \(\ds 12 \, 496\) |
It is interesting to list the prime decomposition of each of the terms in the chain:
\(\ds 12 \, 496\) | \(=\) | \(\ds 2^4 \times 11 \times 71\) | ||||||||||||
\(\ds 14 \, 288\) | \(=\) | \(\ds 2^4 \times 19 \times 47\) | ||||||||||||
\(\ds 15 \, 472\) | \(=\) | \(\ds 2^4 \times 967\) | ||||||||||||
\(\ds 14 \, 536\) | \(=\) | \(\ds 2^3 \times 23 \times 79\) | ||||||||||||
\(\ds 14 \, 264\) | \(=\) | \(\ds 2^3 \times 1783\) |
$\blacksquare$
Also see
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
- 1918: Paul Poulet: $\#$4865 (Interméd. Math. Vol. 25: pp. 100 – 101)
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $28$
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $12,496$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $28$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $12,496$
- Weisstein, Eric W. "Sociable Numbers." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SociableNumbers.html