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. 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)| < x^zb⁺ + o(1), where zb⁺ ∈ (0,1) is an effectively computable constant. In particular, we have that z₁₀⁺ = 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 showed that the results are consistent with our heuristic.