On numbers divisible by the product of their nonzero base $b$ digits

For each integer $b \geq 3$ and every $x \geq 1$, let $\mathcal{N}_{b,0}(x)$ be the set of positive integers $n \leq x$ which are divisible by the product of their nonzero base $b$ digits. We prove bounds of the form $x^{\rho_{b,0} + o(1)} < \#\mathcal{N}_{b,0}(x) < x^{\eta_{b,0} + o(1)}$, as $x \to +\infty$, where $\rho_{b,0}$ and $\eta_{b,0}$ are constants in ${]0,1[}$ depending only on $b$. In particular, we show that $x^{0.526} < \#\mathcal{N}_{10,0}(x) < x^{0.787}$, for all sufficiently large $x$. This improves the bounds $x^{0.495} < \#\mathcal{N}_{10,0}(x) < x^{0.901}$, which were proved by De Koninck and Luca.