Given an ideal $I$, when is $I^nR_P=P^sR_P$ for some $n,s \ge 1$ and some prime ideal $P$

Question

Let $I$ be an ideal of a commutative Noetherian ring $R$. Under what hypotheses can we say that $I^nR_P=P^sR_P$ for some $n,s\ge 1$ and some prime ideal $P$ of $R$?

Note that such a prime ideal $P$ definitely has to contain $I$. If $I=J^r$ is the power of a radical ideal $J$, then also it is clear that $IR_P=P^rR_P$ for every minimal prime $P$ over $I$.

Are there some other reasonable scenarios ?

Thanks in advance.