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BIRS Workshop Lecture Videos

Statistical Analysis of Aliquot Sequences Jacobson, Jr., Michael

Description

Let $s(n) = \sigma(n) - n$ denote the proper sum of divisors function. In his 1976 M.Sc. thesis, Stan Devitt (supervised by Richard Guy) presented theoretical and numerical evidence, using a ``new method of factoring called POLLARD-RHO'', that the average order of $s(n)/n$ in successive iterations of $s(n)$ (Aliquot sequences) is greater than 1. These results seemingly lent support to the Guy/Selfridge Conjecture that there exist unbounded Aliquot sequences. In this talk, we describe the results of a project suggested by Richard in his efforts to provide more evidence in support of the Guy/Selfridge conjecture. In particular, we expand and update Devitt's computations by considering the more-appropriate geometric mean of $s(n)/n$ as opposed to the arithmetic mean considered by Devitt, and greatly extending Devitt's computations using modern factoring algorithms. This is joint work with K. Chum, R. Guy, and A. Mosunov.

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