By Dirichlet's theorem on primes in arithmetic progressions, there are infinitely many odd primes $p$ and $q$ whose sum is divisible by $11$. For such primes $p$ and $q$, you get a solution by taking $r=6(p+q)/11$. So there are infinitely many solutions, and all solutions are of this shape. For Catalan numbers the problem seems to be out of reach.
GH from MO
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