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 ρb ,0 and ηb ,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.