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?

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    $\begingroup$ These are tabulated at oeis.org/A014657 My comment at the earlier question, about the llikelihood of a useful characterization, still holds. $\endgroup$ – Gerry Myerson Aug 12 '19 at 23:59
  • $\begingroup$ Thank you for an answer to the non-prime case. The original question only had the link to the prime case. $\endgroup$ – Edwin Karat Aug 14 '19 at 3:23
  • $\begingroup$ "This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question." @Arturo, that's exactly what Edwin did. This question should not have been closed. I have reopened it. $\endgroup$ – Gerry Myerson Aug 14 '19 at 4:15
  • $\begingroup$ @GerryMyerson: I was not aware my vote would close it by itself. $\endgroup$ – Arturo Magidin Aug 14 '19 at 5:15
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    $\begingroup$ @Arturo, right, I'll edit the title. The question has the group-theory tag, and you have a group-theory gold badge, so one vote from you is all it takes to close (and one vote from me is all it took to reopen, since it has the number-theory tag, and I have a number-theory gold badge). $\endgroup$ – Gerry Myerson Aug 14 '19 at 6:49

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 $n$.


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