Positive integers divisible by the product of their nonzero digits

Abstract

Let ${\mathcal{N}}_0(x)$ denote the set of positive integers $n\le x$ which are divisible by the product of their nonzero digits. In this note, we show that if $x$ is large, then $x^{.495}<\#{\mathcal{N}}_0(x)<x^{.654}$.