Iterating the sum-of-divisors function

Abstract

Let $\sigma^0(n) = n$ and $\sigma^m(n) = \sigma(\sigma^{m-1}(n))$, where $m\ge1$ and $\sigma$ is the sum-of-divisors function. We say that $n$ is $(m,k)$-perfect if $\sigma^m(n) = kn$. We have tabulated all $(2,k)$-perfect numbers up to $10^9$ and all $(3,k)$- and $(4,k)$-perfect numbers up to $2\cdot10^8$. These tables have suggested several conjectures, some of which we prove here. We ask in particular: For any fixed $m\ge1$, are there infinitely many $(m,k)$-perfect numbers? Is every positive integer $(m,k)$-perfect, for sufficiently large $m\ge1$? In this connection, we have obtained the smallest value of $m$ such that $n$ is $(m,k)$-perfect, for $1\le n\le1000$. We also address questions concerning the limiting behaviour of $\sigma^{m+1}(n)/\sigma^m(n)$ and $(\sigma^m(n))^{1/m}$, as $m\to\infty$.