Concatenating in ascending order the 17 smallest palindromic primes with at least two digits gives 1110113115118119131335337338372775778779791992910301, which is prime, so 17 is a term of the sequence.

PROG

(PARI) nextpalprime(n) = my(p=nextprime(n)); while(p!=eval(concat(Vecrev(Str(p)))), p=nextprime(p+1)); p