Counting numbers that are divisible by the product of their digits

Abstract

Let $b\ge 3$ 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 (which consequently must be all nonzero). Let $Z_b\left(x\right)$ 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 $\left|Z_b\right(x\left)\right|<x^{z_b^{+}+o\left(1\right)}$, where $z_b^{+}\in (0,1)$ is an effectively computable constant. In particular, we have that $z_{10}^{+}=0.665\dots$, which improves upon the previous upper bound $\left|Z_{10}\right(x\left)\right|<x^{0.717}$ due to Sanna. Moreover, we prove that $\left|Z_{10}\right(x\left)\right|>x^{0.204}$, which improves upon the previous lower bound $\left|Z_{10}\right(x\left)\right|>x^{0.122}$, due to De Koninck and Luca.

Second, we provide a heuristic suggesting that $\left|Z_b\right(x\left)\right|=x^{z_b+o\left(1\right)}$, where $z_b\in (0,1)$ is an effectively computable constant. In particular, we have that $z_{10}=0.419\dots$.

Third, we provide algorithms to count, respectively enumerate, the elements of $Z_b\left(x\right)$, and we determine their complexities. Implementing one of such counting algorithms, we computed $\left|Z_b\right(x\left)\right|$ for $b=3,\dots ,12$ and large values of x (depending on b), and we show that the results are consistent with our heuristic.