A Divisor-Sum Analogue of the Collatz Map

Abstract

Let $\sigma$ denote the sum-of-divisors function and let $\mathcal{R}$ send an odd integer $n$ to $\sigma (n)$ and an even integer $n$ to $n/2$. We conjecture that every orbit of $\mathcal{R}$ reaches $1$; this implies that there is no odd $2^k$-perfect number for any $k\ge 1$. We prove a pointwise descent estimate for the map $T$ induced by $\mathcal{R}$ on the odd integers, and show that a single application of $T$ divides almost every odd $n$ by $(\log n{)}^{2\log 2-\epsilon }$, for every fixed $\epsilon >0$. The estimate does not iterate, and we determine what is missing.

15 pages; v2: title shortened, paper substantially revised