On the digits of the sum of proper divisors

Abstract

We study several probabilistic questions concerning the digits of $s(n)$, the sum of proper divisors of an integer $n$. In particular, we show that $s(n)$ obeys Benford's law with respect to logarithmic density. Moreover, we show that, for every function $k(x) \rightarrow \infty$, almost all integers $n \leq x$ have every decimal digit occurring among the first $k(x)$ digits and the last $k(x)$ digits of $s(n)$. We also present an upper bound for the number of composite integers $n$ up to $x$ for which $s(n)$ is missing at least one digit in its decimal expansion. This is in contrast with the main result of a recent paper of Benli, Cesana, Dartyge, Dombrowsky, and Thompson, in which the inputs $n$ were not required to be composite. It turns out that the primes make a substantial contribution to the preimage set $s^{-1}(\mathcal{A})$, where $\mathcal{A}$ is a set of integers with missing digits. Our result for composite $n$ shows that the count is much smaller when prime inputs are excluded.