L = { a^{(2^n)} : n>=0 }, is the language regular, context-free but not regular, recursive but not context-free, recursively enumerable?

If regular give a regular expression, if context-free give a context-free grammar, else give a turing machine.

Note: this is "a" to the power of (2 to the power of n), so two powers presented. When n =0, a¹ = a. When n =1, a² = aa. When n =2, a⁴ = aaaa. When n =3, a⁸ = aaaaaaaa. When n =4, a¹⁶ = aaaaaaaaaaaaaaaa.