Let $X$ be a normal variety over $\mathbb{C}$ and $x\in X$ a singular point.
Let $f:Y^{\nu}\to X$ be the normalized blowup at $x\in X$. (i.e. $f$ is a composition of the blowup $Y:=Bl_xX\to X$ and the normalization $Y^{\nu}\to Y$.)
Let $E\subset Y$ be the exceptional divisor of the blowup and $E'$ the pull-back of $E$ to $Y^{\nu}$. Then $-E'$ is $f$-ample, thus $I:=f_*\mathcal{O}_{Y^{\nu}}(-aE')$ generates $\oplus_n f_*\mathcal{O}_{Y^{\nu}}(-naE')$ for some $a>0$.
Q1. Why is $I$ an $m_{X,x}$-primary ideal in $\mathcal{O}_X$?
Q2. What is the blowup of $X$ along $I$? Is it $Y$? $Y^{\nu}$?