Recursively palindromic primes. Are they rare birds?

Question

1. A prime number is a number larger than 1 which only positive divisors are itself and 1.

   Examples: 3,5,11.

2. A number is palindromic in a base $b$ if when written with digits in that basis $d_1d_2\cdots d_n$, $$d_n=d_1\\ d_{n-1} = d_2\\ \cdots$$

For example : the number 191 (in base 10) is a palindromic prime, since it is a prime number and a palindromic number. Now, the sum of the digits, $1+9+1=11$ is also a palindromic prime and $1+1=2$ is also one. So 191 is example of a palindromic prime whose digit sum is a palindromic prime whose digit sum is a $\cdots$ ( and so on).

How many sequences of palindromic primes being preserved by digit sum (all the way down to a 1 digit prime) are there? Infinite or finite? If not infinite, can we calculate how many or give an upper bound?

Answer 1

There are the primes 2, 3, 5, 7, 11.

Then there are palindromic primes with a digit sum of 2, 5, 7, 11. The first ones are 101, 131, 151, 191, 313, 353, 10301, 10501, 11311, 13331, 30103.

Then there are palindromic primes with a digit sum of 101, 131, 151, 191, 313, 353, 10301, 10501, 11311, 13331, 30103. These would be rather large numbers.

Just an opinion: Heuristically, I expect that for any digit sum d, there is only a finite number of palindromic primes with a digit sum d (which is unproven even for the most trivial case d = 2). But also I expect that as d grows, the number of these primes will grow. So I'd expect there to be a rather large number of palindromic primes with a digit sum 30,103, just as an example. So in total, I'd expect the number to be infinite.