# Sociable Chain/Examples/12,496

## 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 $1918$.