Unique Period Prime/Sequence

Sequence

The sequence of unique period primes begins:

$3$, $11$, $37$, $101$, $9091$, $9901$, $333 \, 667$, $909 \, 091$, $99 \, 990 \, 001$, $999 \, 999 \, 000 \, 001$, $9 \, 999 \, 999 \, 900 \, 000 \, 001$, $909 \, 090 \, 909 \, 090 \, 909 \, 091$, $1 \, 111 \, 111 \, 111 \, 111 \, 111 \, 111$, $11 \, 111 \, 111 \, 111 \, 111 \, 111 \, 111 \, 111$, $900 \, 900 \, 900 \, 900 \, 990 \, 990 \, 990 \, 991$, $909 \, 090 \, 909 \, 090 \, 909 \, 090 \, 909 \, 090 \, 909 \, 091$, $\ldots$

This sequence is A040017 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).