Unsinkable numbersMPIM

by Lola Thompson (Utrecht University)

Thursday Jul 16, 2026, 4:30 PM → 6:00 PM Europe/Berlin

MPIM, Vivatsgasse, 7 - Seminar Room (Max Planck Institute for Mathematics)

Number theory lunch seminar 

Let s(n) denote the sum of proper divisors of an integer n. In honour of the Number Theory Boat Seminar, we coin the term "unsinkable numbers" for integers n whose sequence of iterates under s never eventually "sinks" to 1. We will give an overview of what is known about unsinkable numbers as well as some open conjectures.

We will then turn to a different aspect of the sum-of-proper-divisors function: the distribution of the digits of s(n). We will see that s(n) obeys Benford's law and that, asymptotically, almost every value of s(n) contains every decimal digit among both its leading and trailing k(x) digits, where k(x)→∞. We will also discuss recent work showing that values of s(n) missing a decimal digit are surprisingly rare, with prime inputs providing the main contribution, confirming a special case of a conjecture of Erdős, Granville, Pomerance, and Spiro.

This talk is based on joint work with various subsets of the following co-authors: Kübra Benli, Giulia Cesana, Cécile Dartyge, Charlotte Dombrowsky, and Paul Pollack.