Number Theory Seminar

Anton Mosunov, Department of Pure Mathematics, University of Waterloo

“What do we know about aliquot sequences? (in honor of Richard Guy’s 100th birthday) ”

Let s(n) denote the sum of the proper divisors of n and sk(n) denote the k-th iterate of s. An aliquot sequence starting at n is a sequence of the form n,s(n),s2(n),s3(n), and so on. There has been a lot of discussion about the convergence of these sequences. On the one hand, the Catalan- Dickson conjecture (1913) states that all aliquot sequences converge. On the other hand, there exists a heuristic counter-argument due to Guy and Selfridge (1975), according to which the divergence occurs for almost all aliquot sequences starting with an even number. In this talk, we intend to summarize all theoretical observations about aliquot sequences and present a series of computational results on the heuristics of Guy and Selfridge. This is joint work with Kevin Chum, Richard K. Guy and Michael J. Jacobson, Jr. (University of Calgary).

MC 5479