Counting numbers that are divisible by the product of their digits

Abstract

Let $b \geq 3$ be a positive integer. A natural number is said to be a base-$b$ Zuckerman number if it is divisible by the product of its base-$b$ digits. Let $\mathcal{Z}_b(x)$ be the set of base-$b$ Zuckerman numbers that do not exceed $x$, and assume that $x \to +\infty$. First, we prove an upper bound of the form $|\mathcal{Z}_b(x)| < x^{z_b^{+} + o(1)}$, where $z_b^{+} \in (0,1)$ is an effectively computable constant. In particular, we have that $z_{10}^+ = 0.665{\scriptstyle\ldots}$, which improves upon the previous upper bound $|\mathcal{Z}_{10}(x)| < x^{0.717}$ due to Sanna. Moreover, we prove that $|\mathcal{Z}_{10}(x)| > x^{0.204}$, which improves upon the previous lower bound $|\mathcal{Z}_{10}(x)| > x^{0.122}$, due to De Koninck and Luca. Second, we provide a heuristic suggesting that $|\mathcal{Z}_b(x)| = x^{z_b + o(1)}$, where $z_b \in (0,1)$ is an effectively computable constant. In particular, we have that $z_{10} = 0.419{\scriptstyle\ldots}$. Third, we provide algorithms to count, respectively enumerate, the elements of $\mathcal{Z}_b(x)$, and we determine their complexities. Implementing one of such counting algorithms, we computed $|\mathcal{Z}_b(x)|$ for $b=3,\dots,12$ and large values of $x$ (depending on $b$), and we showed that the results are consistent with our heuristic.