Let $\pi : Z = \widetilde{\mathbb{P}^{3}} \longrightarrow \mathbb{P}^{3}$ be the blowing up of $\mathbb{P}^{3}$ along an irreducible non-degenerate smooth curve $\mathcal{C}$ of degree $d$. According to the question (Direct Image by a Blow up) and answer we have $$\pi_{*}\mathcal{O}_{Z}(-nE) = I_{\mathcal{C}/ \mathbb{P}^{3}}^{n}$$
In the following papper : On The Castelnuovo-Mumford Regularity Of Product Of Ideal sheaves, Jessica Sidman (arXiv:math/0110184) we have
Proposition 1.7 : Suppose that $\mathcal{I}$ and $\mathcal{J}$ are ideal sheaves on $\mathbb{P}^{n}$ with regularities $m_{1}$ and $m_{2}$, respectively. Suppose also that zeros of $\mathcal{I}$ and $\mathcal{J}$ intersect in a set of dimension at most one. Then $\mathcal{I} \cdot \mathcal{J}$ is $(m_{1} + m_{2})$-regular.
Now, according to proposition 1.8.46 of Positivity in Algebraic Geometry I, we have $\mathcal{I}_{\mathcal{C}}$ is $(d - 1)$-regular.
Suppose that proposition 1.7 applies to a finite number of ideal sheaves.
Making $\mathcal{I} = \mathcal{I}_{\mathcal{C}} = \mathcal{J}$ in the proposition 1.7 we can conclude that $$\mathcal{I}_{\mathcal{C}}^{n} = \underbrace{\mathcal{I}_{\mathcal{C}} \cdot \mathcal{I}_{\mathcal{C}} \cdots \cdot \mathcal{I}_{\mathcal{C}}}_{n-\text{times}}$$ is $$n \bigl(d - 1 \bigr)-\text{regular}?$$
Thank you.
