imc

We're given $(130n{)}^{50}>n^{100}>2^{200}$, so $\sqrt[50]{(130n{)}^{50}}>\sqrt[50]{n^{100}}>\sqrt[50]{2^{200}}$ (because all terms are positive) and thus $130n>n^2>2^4$ $130n>n^2>16$ Solving each part separately: $n^2>16⟹n>4$ $130n>n^2⟹130>n$ So $4<n<130$. Therefore the answer is the number of positive integers over the interval $(4,130)$ which is $129-5+1=\boxed{125}$