p-adic valuations and k-regular sequences

A sequence is said to be k-automatic if the imageth term of this sequence is generated by a finite state machine with n in base k as input. Regular sequences were first defined by Allouche and Shallit as a generalization of automatic sequences. Given a prime p and a polynomial image, we consider the sequence image, where image is the p-adic valuation. We show that this sequence is p-regular if and only if image factors into a product of polynomials, one of which has no roots in image, the other which factors into linear polynomials over image. This answers a question of Allouche and Shallit.