Imagine the sequence of palindromic numbers where each term is defined as the smallest square-free palindromic number with no other prime factors but the n distinct palindromic prime factors. The sequence begins:
- $2$
- $6=2\cdot3$
- $66=2\cdot3\cdot11$
- $6666=2\cdot3\cdot11\cdot101$
What is the next term? Does the next term exist? I know that if there are infinitely many primes of the form $100000....000001$,then this sequence is definitely infinite, but it doesn't mean the infinitude of this sequence depends entirely on the infinitude of primes of the form $100000....000001$. I've checked palindromes up to $10^{10}$ now. So,what is the next term ?
I am not sure I understood the definition of the sequence, but $$ 334\,826\,628\,433=11\cdot101\cdot353\cdot919\cdot929 $$ seems to be the next term.