Question 138
Let $n \gt 4$ be a positive integer which is divisible by $4$. We denote by $A_n$ the sum of all odd positive divisors of $n$. We denote by $B_n$ the sum of all even positive divisors of $n$, with the exclusion of $n$. Find the smallest possible value of $f(n) = B_n - 2A_n$. For which values of the positive integer $n$ is this minimal value obtained?