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.
Categories: math.NT