aliquot sequence

For a given $m$, define the recurrence relation $a_1=m$, $a_n=\sigma (a_{n-1})-a_{n-1}$, where $\sigma (x)$ is the sum of divisors function. $a$ is then the aliquot sequence of $m$.

If $m$ is an amicable number, its aliquot sequence is periodic, alternating between the abundant and deficient member of the amicable pair. For a prime number $p$, its aliquot sequence is $p,1,0$. In other cases, the aliquot sequence reaches a fixed point upon 0, or on a perfect number.