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Probably $p$ is the projection $Bl \mathbb{P}^n \to \mathbb{P}^n$ and $i$ the reverse morphism, the inclusion $\mathbb{P}^n \hookrightarrow Bl\mathbb{P}^n$.
Can you give an example of an $a_n$ that satisfies the given condition? I only see something like $2^{2^n}$ which is not something one encounter on a daily basis
