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Conditions:

n such that $\ Ord_n(2) \mid n-1 $ and $\ Ord_n(2) - 1 = 2^x,n \in >2\mathbb{N}+1,\ x \in \mathbb{Z}_{\geq 0}$.

I check up to 1e7 :

3,7,31,73,6151,57457,131071,599479,1572871,2252951,5242921

Numbers matching the condition are all primes.Are there composite numbers matching the conditions?

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Yes: $n = 1227133513 = 23 \cdot 89 \cdot 599479$ appears to be the smallest, having $\text{ord}_n(2) = 33$. However, there seems to be no further examples up to $10^{20}$. It's not surprising that they are very rare, since pseudoprimes are already much rarer than the primes.

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