A Note on the Distribution of k-Full Integers in Arithmetic Progressions

Abstract

A k−full integer (k ≥ 2) is a positive integer n such that p^k divides n whenever p is a prime divisor of n. Let N_k be the set of such integers. For a real x ≥ 1, we present an asymptotic formula for the number of natural numbers {n ≤ x, n ∈ N_k} such that the sum of their digits, s_g(n), in base g ≥ 2 satisfies s_g(n) ≡ a mod b, where a ∈ Z and b ≥ 2.

Keywords: Exponential sums, K-full integers, Sum of digits function.

2020 Mathematics Subject Classification. 11A63, 11L03, 11N69