Recursive style

Let $A_n$ be a recursive formula that satisfies

$A_1=k, \;\; A_2 \neq 2, \;\; A_n=D(A_{n-1}) \;\;$ where $k$ is a positive integer and $D(t)$ is the number of divisors of $t$ .

Does $\{A\}$ contain a perfect square?

Bonus: prove your answer! $:)$

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This problem is not original.