The number $r$ with the continued fraction $[0,2,3,5,7,11,13,17,19,23,29,\cdots]$ (The elements of the sequence are the primes in increasing order) begins with the digits $$0.43233208718590286890925379324199996370511$$
So, already the digits $31$ to $34$ are consecutive nines which is unusually early. This is similar to the "Feynman-point" of $\pi$
Is this just a coincide, or is there an explanation based on the convergents or best approximations of $r$ ?