ILL 1987 Problem 71

To every natural number $k, k \geq 2$ , there corresponds a sequence $a_n(k)$ according to the following rule: $$a_0 = k, a_n = \tau(a_{n-1}) \forall n \geq 1,$$ in which $\tau(a)$ is the number of different divisors of $a$ . Find all $k$ for which the sequence $a_n(k)$ does not contain the square of an integer.