Proving that L is not regular by showing that $\equiv_L$ has infinite index
Let me prove a stronger claim: every two words over $\{a\}$ are inequivalent with respect to your language. What this means is that for every $i \neq j$, there is a word $w \in a^*$ such that either $a^i w \in L$ and $a^j w \notin L$, or vice versa. Since $w \in a^*$, in fact $w = a^k$ for some $k$, and so we need to prove the following:


  If $i \neq j$ are natural numbers then there exists a natural number $k$ such that exactly one of $i+k,j+k$ is a power of 3.


Let $a_i$ be the smallest power of 3 which is at least $i$, and define $a_j$ similarly. If $a_i \neq a_j$, say $a_i &lt; a_j$, then we can take $k = a_i$. So suppose $a_i = a_j = a$. Without loss of generality, $i &lt; j$, and so $i+a &lt; j+a$. The next power of 3 after $i+a$ is $3(i+a)$, and after $j+a$ is $3(j+a)$. Let $k = 2i+3a$. Then $i + k$ is a power of 3 by definition, but $j+a &lt; j + k &lt; 3(j+a)$ is not a power of 3.