  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.
