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.Comment: 10 pages, to appear in Mathematics of Computatio