Question 39
For a positive integer $n$, we define $\alpha(n)$ as the average of all positive divisors of $n$, and $\beta(n)$ as the average of all positive integers $k \leq n$ such that $\gcd(k, n)=1$.
Find all positive integers $n$ for which $\alpha(n)=\beta(n)$.
Find all positive integers $n$ for which $\alpha(n)=\beta(n)$.