Finite connected components of the aliquot graph

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.