Question 112
We call the permutation $\pi$ of $\{1,2, ..., n\}$ *consistent* if the set $\{\pi(k) - k \mid k = 1,2, ..., n\}$ has 2 members. Prove that the total number of *consistent* permutations is $\sigma(n) - \tau(n)$, where $\sigma(n)$ is the sum of the positive divisors of $n$ and $\tau(n)$ is the number of positive divisors of $n$.