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

Abstract

For each integer $b\ge 3$ and every $x\ge 1$, let ${\mathcal{N}}_{b,0}(x)$ be the set of positive integers $n\le x$ 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.

Cited by in corpus (1)

• Additive bases and Niven numbers