Finite connected components of the aliquot graph

TL;DR

This paper classifies all finite connected components of the aliquot graph for numbers below certain large thresholds, assuming a strong Goldbach conjecture, and introduces a fast algorithm for inverse sum-of-proper-divisors computations.

Contribution

It provides a comprehensive classification of finite components of the aliquot graph under specific bounds, assuming a strong Goldbach conjecture, and develops a new efficient algorithm for inverse divisor sum calculations.

Findings

All finite connected components below 10^9 are identified.

The algorithm efficiently computes inverse images under the sum-of-proper-divisors function.

Results depend on a strong form of the Goldbach conjecture.

Abstract

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.