Investigation of Symbol Minimal Context-Free Grammars for the Language $a^n$
First, let me add some more to the known results section, specific to your question.


$B$ must occur in another RHS than $A\rightarrow aB$, or we could just copy the rule that has $B$ on the LHS instead of $B$ to the above rule to get a smaller system.
In any other RHS that has $B$, there can be no terminal symbol because instead of $aB$ we could write $A$.
$A$ must occur at least $4$ times on the RHS, otherwise, we could just write $aB$ instead of it.


This already shows that $n$ will be quite large that satisfies these. However, I think such an $n$ might exist. Below I give a set of rules for $n=93$ with $28$ symbols and I challenge all to make a smaller system! (If the challenge is too easy, generalize it in the obvious way...)

$S\rightarrow AAAACCC$

$A\rightarrow aB$

$B\rightarrow aaa$

$C\rightarrow DDD$

$D\rightarrow BBB$

Update

Well, this is not optimal for any $n$ as shown by Peter in the comment...