What is the next such palindrome

Consider the sequence of palindromic numbers where each term is defined as the smallest palindromic number with exactly k distinct palindromic prime factors (to avoid ambiguity, here I mean a palindromic number whose prime factors are ALL also palindromic numbers.). The sequence begins: $2$,$6$,$66$,$6666$:

• $2$
• $6$=$2$.$3$
• $66$=$2$.$3$.$11$
• $6666$=$2$.$3$.$11$.$101$

What is the next term after $6666$,does the next term exist ?. I have checked palindromes up to $10^9$( But I know that if there are infinitely many primes of the form $10000....00001$,then this sequence is absolutely infinite.). So what is the next term ?

• Where does this come from? It looks like a contest, maybe ProjectEuler? – Jean-Claude Arbaut Oct 24 '15 at 11:01
• @Jean-ClaudeArbaut,no, it just come to my head : D – Orange Diamond Oct 24 '15 at 11:04

$$a_0=6,\quad a_{n+1}=a_n\cdot (10^{2^{n-1}}+1), \quad n\ge3$$ are of numbers with $2^n$ repeated 6's.
• I mean any palindrome, not just restricted to string of digit $6$ – Orange Diamond Oct 24 '15 at 10:57