On $k$-regularity of sequences of valuations and last nonzero digits

Abstract

Let $b\ge 2$ be an integer base with prime factors $p_1,\dots ,p_s$. In this paper we study sequences of "$b$-adic valuations" and last nonzero digits in $b$-adic expansions of the values $f(n)=(f_1(n),\dots ,f_s(n))$, where each $f_i$ is a $p_i$-adic analytic function. We give a complete classification concerning $k$-regularity of these sequences, which generalizes a result for $b$ prime obtained by Shu and Yao. As an application, we strengthen a theorem by Murru and Sanna on $b$-adic valuations of Lucas sequences of the first kind. Moreover, we derive a method to determine precisely which terms of these sequences can be represented by certain ternary quadratic forms.