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

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.

Mathematics Subject Classification (2010): Primary: 11A63; Secondary: 11N25.