p-adic valuations and k-regular sequences

A sequence is said to be k-automatic   if the nnth 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 f(x)∈Qp[x]f(x)∈Qp[x], we consider the sequence {vp(f(n))}n=0∞, where vpvp is the p-adic valuation. We show that this sequence is p  -regular if and only if f(x)f(x) factors into a product of polynomials, one of which has no roots in ZpZp, the other which factors into linear polynomials over QQ. This answers a question of Allouche and Shallit.