Counting numbers that are divisible by the product of their digits

Let b ≥ 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 Zb(x) be the set of base-b Zuckerman numbers that do not exceed x, and assume that x → +∞. First, we prove an upper bound of the form |Zb(x)| < xz+ b +o(1), where z+ b ∈ (0, 1) is an effectively computable constant. In particular, we have that z+ 10 = 0.665..., which improves upon the previous upper bound |Z10(x)| < x0.717 due to Sanna. Moreover, we prove that |Z10(x)| > x0.204, which improves upon the previous lower bound |Z10(x)| > x0.122, due to De Koninck and Luca. Second, we provide a heuristic suggesting that |Zb(x)| = xzb+o(1), where zb ∈ (0, 1) is an effectively computable constant. In particular, we have that z10 = 0.419.... Third, we provide algorithms to count, respectively enumerate, the elements of Zb(x), and we determine their complexities. Implementing one of such counting algorithms, we computed |Zb(x)| for b = 3,..., 12 and large values of x (depending on b), and we show that the results are consistent with our heuristic