On the Euler function of the Catalan numbers

We study the solutions of the equation ϕ(Cm)/ϕ(Cn)=r, where r is a fixed rational number, Ck is the kth Catalan number and ϕ is the Euler function. We note that the number r=4 is special for this problem and for it we construct solutions (m,n) to the above equation which are related to primes p such that 2p−1 or 4p−3 is also prime.