For a definition, see the wikipedia page: http://en.wikipedia.org/wiki/Strictly_non-palindromic_number

So according to the wikipedia page, under properties, all strictly non-palindromic numbers with three exceptions are primes.

So I have some questions about these numbers. First, what is their relative density in the primes? Does the sum of their reciprocals converge? If not, then do they contain arbitrarily long arithmetic progressions? Are they known to be an additive basis of any order?

Almost all palindromes are composite. Math. Res. Lett. 11 (2004), no. 5-6, 853–868. Maybe this could be used to prove that almost all prime numbers are non-palindromic (at least in a given base). On the other hand I'm not sure it has been proved (or even is easy to show) that there are infinitely many strictly non-palindromic numbers... – François Brunault May 31 '11 at 8:54