Fall 2021 AMC 10 B

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 $2+1+1+3+1=8$.