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Mathematics of Computation

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Finite connected components of the aliquot graph

Author: Andrew R. Booker
Journal: Math. Comp. 87 (2018), 2891-2902
MSC (2010): Primary 11A25
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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Additional Information

Andrew R. Booker
Affiliation: School of Mathematics, University of Bristol, University Walk, Bristol, BS8 1TW, United Kingdom

Received by editor(s): October 24, 2016
Received by editor(s) in revised form: February 15, 2017, May 3, 2017, and May 27, 2017
Published electronically: February 20, 2018
Additional Notes: The author was partially supported by EPSRC Grant EP/K034383/1.
Article copyright: © Copyright 2018 American Mathematical Society

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