Sociable Chain/Examples/12,496

Example of Sociable Chain
The longest known sociable chain, at time of writing ($9$th March $2017$) is of order $28$.

Its smallest element is $12 \, 496$.

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.

Throughout, the result Sigma of Integer is to be used:
 * $\displaystyle \sigma \left({n}\right) = \prod_{1 \mathop \le i \mathop \le r} \frac {p_i^{k_i + 1} - 1} {p_i - 1}$

where $\displaystyle n = \prod_{1 \mathop \le i \mathop \le r} p_i^{k_i}$ is the prime decomposition of $n$.

Thus: