  For integers $k \geq 2$ and $n \neq 0$, let $v_k(n)$ denotes the greatest
nonnegative integer $e$ such that $k^e$ divides $n$. Moreover, let $u_n$ be a
nondegenerate Lucas sequence satisfying $u_0 = 0$, $u_1 = 1$, and $u_{n + 2} =
a u_{n + 1} + b u_n$, for some integers $a$ and $b$. Shu and Yao showed that
for any prime number $p$ the sequence $v_p(u_{n + 1})$ is $p$-regular, while
Medina and Rowland found the rank of $v_p(F_{n + 1})$, where $F_n$ is the
$n$-th Fibonacci number. We prove that if $k$ and $b$ are relatively prime then
$v_k(u_{n + 1})$ is a $k$-regular sequence, and for $k$ a prime number we also
determine its rank. Furthermore, as an intermediate result, we give explicit
formulas for $v_k(u_n)$, generalizing a previous theorem of Sanna concerning
$p$-adic valuations of Lucas sequences.
