2021 AIME II Problem 15

Let f(n) and g(n) be functions satisfying f(n) = \begin{cases} \sqrt{n} & \text{ if } \sqrt{n} \text{ is an integer}\\ 1 + f(n+1) & \text{ otherwise} \end{cases} and g(n) = \begin{cases}\sqrt{n} & \text{ if } \sqrt{n} \text{ is an integer}\\ 2 + g(n+2) & \text{ otherwise} \end{cases} for positive integers n. Find the least positive integer n such that \tfrac{f(n)}{g(n)} = \tfrac{4}{7}.

Leading zeroes must be inputted, so if your answer is 34, then input 034

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.