Every palindrome with an even number of digits is divisible by 11, so 11 is the only member of the sequence with an even number of digits. - David Wasserman, Sep 09 2004

This holds in any number base A006093(n), n>1. - Lekraj Beedassy, Mar 07 2005 and Dec 06 2009

The log-log plot shows the fairly regular structure of these numbers. - T. D. Noe, Jul 09 2013

Conjecture: The only primes with palindromic prime indices that are palindromic primes themselves are 3, 5 and 11. Tested for the primes with the first 8000000 palindromic prime indices. - Ivan N. Ianakiev, Oct 10 2014

Banks, Hart, and Sakata derive a nontrivial upper bound for the number of prime palindromes n <= x as x -> oo. It follows that almost all palindromes are composite. The results hold in any base. The authors use Weil's bound for Kloosterman sums. - Jonathan Sondow, Jan 02 2018