For which odd positive integer $n$, is it true that $−1$ is not a positive power of $2$ modulo $n$ i.e. $[−1]\ne [2^k],\forall k>0$ in $\mathbf{Z}_n$ ?
This question was asked in For which odd positive integer $n$ , is it true that $-1$ is not a positive power of $2$ modulo $n$?
There, the following sequence was referenced: https://oeis.org/A091317
However, the sequences given only cover $n$ prime. What if $n$ is not prime?
This is not completely trivial, as $n=15$ does not have $-1$ as a positive power modulo $n$, even though $n=3$ and $n=5$ do. That is, $3$ divides $2^n + 1$ for some $n$, and $5$ divides $2^n + 1$ for some $n$, but $15$ does not divide $2^n + 1$ for some $n$. At the same time, $33$ does divide $2^n + 1$ for some $n$ (as do $3$ and $11$, obviously).
Is there a way to apply this sequence to composites and prime powers?
As I mention in my comment, the sequence is tabulated at https://oeis.org/A014657
It seems unlikely to me that anyone can say very much that's useful about those values of $n$.