Finite connected components of the aliquot graph

Booker, Andrew

doi:10.1090/mcom/3299

Finite connected components of the aliquot graph
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by Andrew R. Booker;

Math. Comp. 87 (2018), 2891-2902

DOI: https://doi.org/10.1090/mcom/3299

Published electronically: February 20, 2018

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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.

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