A069866 Primes in which repeatedly deleting the most significant digit then the least significant digit gives a prime at every step until a single digit prime remains.

2, 3, 5, 7, 13, 17, 23, 37, 43, 47, 53, 67, 73, 83, 97, 131, 137, 173, 179, 223, 229, 271, 331, 337, 353, 359, 373, 379, 431, 479, 523, 571, 631, 653, 659, 673, 773, 823, 829, 853, 859, 929, 937, 953, 971, 1031, 1373, 1433, 1439, 1733, 2029, 2053, 2131, 2137