On numbers divisible by the product of their nonzero base $b$ digits

TL;DR

This paper establishes new bounds on the number of integers up to x that are divisible by the product of their nonzero base b digits, refining previous estimates especially for base 10.

Contribution

It provides improved asymptotic bounds for the count of such integers, introducing constants that depend only on the base and tightening previous bounds.

Findings

Bounds of the form x^{ ho_{b,0}+o(1)} and x^{ heta_{b,0}+o(1)} for the count

Explicit bounds for base 10: between x^{0.526} and x^{0.787}

Improved previous bounds from x^{0.495} to x^{0.901} for base 10

Abstract

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.