Arithmetic properties of positive integers with fixed digit sum

Abstract

In this paper, we look at various arithmetic properties of the set of those positive integers $n$ whose sum of digits in a fixed base $b>1$ is a fixed positive integers $s$. For example, we prove that such integers can have many prime factors, that they are not very smooth, and that most such integers have a large prime factor dividing the value of their Euler $\varphi$ function.