Is there a Poulet number $n$ with this condition: $◎(n)=\frac{n+1}{2^x}$ or $\ ◎(n)=\frac{n-1}{2^x}, \ x \in \mathbb{N}_{\gt 0}$? (Recall that a Poulet number is a composite $n$ such that $2^n−2$ is divisible by $n$.)
Here $◎(n)$ is called the cycle length of $n$. For an example satisfying this condition: $◎(29)=14=\frac{29-1}{2^1}$.
For detail of the cycle length of $n$ see: How to prove these two ways give the same numbers?.
Known Composite numbers satisfying this condition are:
92673, 143713, 3579553, 4110529, 28688897, 127017857, 141127681, 157648097, 162101441, 212999489, 393300097, 663414881
See http://oeis.org/A225890 (all aren't Poulet number)
If non-existence of such numbers proved imply this fact: n must be prime such that $\ \frac{n-1}{ord_n 2}=2^x\ $ and $◎(n)=\frac{n \pm 1}{2^y},\ n \in \mathbb{Z^+} ,\ x \in \mathbb{Z}_{\geq 0},\ \ y \in \mathbb{Z}_{\gt 0}$.
The form $\ \frac{n-1}{ord_n 2}=2^x\ $ see: https://mathoverflow.net/questions/168045/are-all-counterexamples-of-oeis-a226181-both-poulet-numbers-and-proth-numbers.
