A prime such that its decimal period's length is shared with no other prime is called a unique prime (as defined by Yates in 1980).
3, 11, 37, and 101 are unique primes, since they are the only primes with periods one ( 1/3=.3333), two (1/11=.090909... ), three and four respectively.

List the first 15 unique primes, explaining how you obtained them.

Copying a list from OEIS does not qualify as a solution, getting information from any source is encouraged.