Counting Primes whose Sum of Digits is Prime

Motivated by recent work of Drmota, Mauduit and Rivat, we discuss the possibility of counting the number of primes up to x whose sum of digits is also prime. We show that, although this is not possible unless we assume a hypothesis on the distribution of primes stronger than what is implied by the Riemann hypothesis, we can establish a Mertens-type result. That is, we obtain a formula for the number of such primes p up to x weighted with a factor 1/p. Indeed, we can iterate the method and count primes with the sum of digits a prime whose sum of digits is a prime, and so on.