Research Article

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

Published in: Quaestiones Mathematicae
Volume 43 , issue 11 , 2020 , pages: 1563–1571

DOI: 10.2989/16073606.2019.1637956

Author(s): Carlo Sanna , Italy

Keywords: Primary: 11A63 , Secondary: 11N25 , Primary: 11A63 , Secondary: 11N25

Abstract

For each integer b ≥ 3 and every x ≥ 1, let _b _,0(x) be the set of positive integers n ≤ x which are divisible by the product of their nonzero base b digits. We prove bounds of the form x ^ρb,0+o(1) < # _b _,0(x) < x ^ηb,0+o(1), as x → +∞, where ρ_b _,0 and η_b _,0 are constants in ]0, 1[ depending only on b. In particular, we show that x ^0.526 < # _10,0(x) < x ^0.787, for all suﬃciently large x. This improves the bounds x ^0.495 < # _10,0(x) < x ^0.901, which were proved by De Koninck and Luca.

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