  Conditional on a strong form of the Goldbach conjecture, we determine all
finite connected components of the aliquot graph containing a number less than
$10^9$, as well as those containing an amicable pair below $10^{14}$ or one of
the known perfect or sociable cycles below $10^{17}$. Along the way we develop
a fast algorithm for computing the inverse image of an even number under the
sum-of-proper-divisors function.
