Unique Period Prime/Examples

Examples of Unique Primes
In base $10$, the prime numbers $3$, $11$, $37$, and $101$ are the only ones with periods $1$, $2$, $3$ and $4$ respectively.

Hence they are unique primes.

But:
 * $41$ and $271$ both have period $5$
 * $7$ and $13$ both have period $6$
 * $239$ and $4649$ both have period $7$
 * $73$ and $137$ both have period $8$.

Thus these primes are not unique primes.