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

For each integer b ≥ 3 and every x ≥ 1, let (Figure presented.) 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 xρjavax.xml.bind.JAXBElement@43ace5c8+o(1) < # (Figure presented.) b,0(x) < xηjavax.xml.bind.JAXBElement@103fdc7f+o(1), as x → +∞, where ρb,0 and ηb,0 are constants in ]0, 1[ depending only on b. In particular, we show that x0.526 < # (Figure presented.) 10,0(x) < x0.787, for all sufficiently large x. This improves the bounds x0.495 < # (Figure presented.) 10,0(x) < x0.901, which were proved by De Koninck and Luca.