If $f$ is any monic polynomial/$\mathbb{Z}$ with non-zero constant coefficient. I wish to study the quantities

$$t_n=\sum_{i}\theta_i^n\in\mathbb{Z}$$

where $(\theta_i)_{i=1}^{d}$ are the roots of $f$ counted with multiplicity.

The main question I am interested is finding all the primes $p$ such that $p\mid t_n$ for all large enough $n$. I have proved that this can only happen in the case that either $p\mid c_n$  $\forall$ $n$ where $c_n$ are the non-leading coefficients of $f$, or $p\mid t_n$ $\forall$ $n\geq0$.

The condition that $p\mid c_n$ is easy to check, so my interest has turned to some algebraic interpretation of the condition that $p\mid t_n$ for all $n$. My current idea is to let $K=\mathbb{Q}(\theta_i)_{i=1}^{d}$ be the field attached to $f$, and analyse the ideal $I$ of $\mathcal{O}_K$ generated by $(\theta_i)_{i=1}^{d}$. My idea is that it seems as if $p\mid t_n$ $\forall$ $n$ implies that any element of $I \cap \mathbb{Z}$ will be a multiple of $p$. The converse is obvious.

Is the above observation correct? If so, is there some known property of such primes (i.e. $I\cap \mathbb{Z} \subset p\mathbb{Z}$ iff $p$ ramifies in $\mathcal{O}_K$)?