CN-TST 2014 Problem 8

Given a fixed positive integer $a\geq 9$ . Prove: There exist finitely many positive integers $n$ , satisfying: (1) $\tau (n)=a$ (2) $n|\phi (n)+\sigma (n)$ Note: For positive integer $n$ , $\tau (n)$ is the number of positive divisors of $n$ , $\phi (n)$ is the number of positive integers $\leq n$ and relatively prime with $n$ , $\sigma (n)$ is the sum of positive divisors of $n$ .