I think $L$ is not regular. Here's an idea: let $k$ be the number of states of a minimum DFA that accepts $L$. Since the alphabet has one symbol only, this DFA must be a cycle (why?). For an appropriate $n$ there exists $m$ such that $f(m+1)-f(m)>k$, which is a contradiction, because there would be some $c$ between $f(m)$ and $f(m+1)$ such that $a^c$ belongs to $L$ (why?). However, $c$ can not be the image of an integer by $f$ (why?).
I leave the details to you, first because given your only other question in this website I suspect that I'm doing your homework, and second because I gave you enough hints already :) Good luck!
Update: I'm a new user, and this was my first answer. I realize now that the website is for research-level questions only. That said, this question looked like homework to me, so maybe somebody will want to close the topic.