HMMT November 2023 U36

Isabella writes the expression $\sqrt{d}$ for each positive integer $d$ not exceeding 8 ! on the board. Seeing that these expressions might not be worth points on HMMT, Vidur simplifies each expression to the form $a \sqrt{b}$, where $a$ and $b$ are integers such that $b$ is not divisible by the square of a prime number. (For example, $\sqrt{20}, \sqrt{16}$, and $\sqrt{6}$ simplify to $2 \sqrt{5}, 4 \sqrt{1}$, and $1 \sqrt{6}$, respectively.) Compute the sum of $a+b$ across all expressions that Vidur writes. Submit a positive real number $A$. If the correct answer is $C$ and your answer is $A$, you get $\max \left(0,\left\lceil 20\left(1-|\log (A / C)|^{1 / 5}\right)\right\rceil\right)$ points.