OEIS A226181:

3, 5, 7, 11, 13, 17, 19, 23, 29, 37, 41, 47, 53, 59, 61, 67, 71, 73, 79,  
83, 89, 97, 101, 103, 107, 113, 131, 137, 139, 149, 163, ...

Primes $p$ such that $p-1$ divided by the period of the binary expansion of $1/p$ equals $2^x$ for some nonnegative integer $x$. Composite numbers matching the conditions below $10^8$ are:

 12801, 348161, 3225601  

Poulet n. and Proth n. :

Poulet n. : A composite n such that $2^n - 2$ is divisible by $n.\ $ Proth n. : A number of the form $k ⋅ 2^n + 1$, with $k$ odd, $n > 0$, and $2n > k$.

First numbers that belong to both sets:

1729, 4033, 8321, 12801, 65281, 130561, 348161, 3225601, 8355841,
8384513, 16773121, 40280065, 104988673, 2147418113,
4294901761, 4294967297, 53282340865, 68719214593, 137439477761.

Here's the question: does all counterexamples of OEIS A226181 are both Poulet numbers and proth numbers?