, 2021

Because $10^r$ is written as a $1$ followed by $r$ zeros, the $r$th root of any number smaller than $10^r$ when written as a decimal has only one digit to the left of the decimal point. Extending this reasoning, the leading digit of the $r$th root of $N$ is the same as the leading digit of the $r$th root of the integer $n$ that has as digits a number of $7$s equal to the remainder when $313$ is divided by $r$. For example, the fifth root of $N$ has the same leading digit as the fifth root of $777$, because the remainder of $313$ when divided by $5$ is $3$. Because $3^5 = 243$ and $4^5 = 1024$, the leading digit is $3$.

Using this methodology, it follows that $f(2) = 2$, $f(3) = 1$, $f(4) = 1$, $f(5) = 3$, and $f(6) = 1$. The requested sum is $2 + 1 + 1 + 3 + 1 = 8$.