For each positive integer n, let a_{n} be the number of permutations \tau of \{1,2, \ldots, n\} such that \tau(\tau(\tau(x)))=x for x=1,2, \ldots, n. The first few values are
a_{1}=1, a_{2}=1, a_{3}=3, a_{4}=9
Prove that 3^{334} divides a_{2001}.
(A permutation of \{1,2, \ldots, n\} is a rearrangement of the numbers \{1,2, \ldots, n\}, or equivalently, a one-to-one and onto function from \{1,2, \ldots, n\} to \{1,2, \ldots, n\}. For example, one permutation of \{1,2,3\} is the rearrangement \{2,1,3\}, which is equivalent to the function \sigma:\{1,2,3\} \rightarrow\{1,2,3\} defined by \sigma(1)=2, \sigma(2)=1, \sigma(3)=3.)