On the Skolem Problem and Prime Powers

The Skolem Problem asks, given a linear recurrence sequence $(u_n)$, whether there exists $n\in \mathbb{N}$ such that $u_n=0$. In this paper we consider the following specialisation of the problem: given in addition $c\in \mathbb{N}$, determine whether there exists $n\in \mathbb{N}$ of the form $n=l{p}^k$, with $k,l\le c$ and $p$ any prime number, such that $u_n=0$.