2004 IMO Shortlist S13

Let $\tau(n)$ denote the number of positive divisors of the positive integer $n$ . Prove that there exist infinitely many positive integers $a$ such that the equation $\tau(an)=n$ does not have a positive integer solution $n$ .

设 $\tau(n)$ 表示正整数 $n$ 的正因数数。证明存在无穷多个正整数 $a$ 使得方程 $\tau(an)=n$ 没有正整数解 $n$ 。