Seminario: "On numbers divisible by the product of their digits" - Carlo Sanna
Combinatorial Geometry and Number Theory
Bernoulli Center, EPFL Campus, Lausanne, August 26-30, 2024
https://ilariaviglino.wixsite.com/combgeomepfl
Abstract. Let b >= 3 be a positive integer. A natural number is said to be a (base-b) Zuckerman number if it is divisible by the product of its (base-b) digits. In this talk we will show new results on the counting function Z_b(x) of Zuckerman numbers. First, we prove new upper and lower bounds for Z_b(x), improving upon previous results of De Koninck–Luca and Sanna. Second, we provide a heuristic suggesting an asymptotic formula for Z_b(x). Third, we provide algorithms to compute Z_b(x), and we determine their complexities. By implementing one of such algorithms, we computed Z_b(x) for b=3,...,12 and large values of x, and we showed that the results are consistent with our heuristic.
This is a joint work with Qizheng He.