Gcd and Catalan numbers

Question

Let $C_n= \binom{2n}{n}/(n+1)$ be the catalan sequence and $b_r$ the sequence such that $\gcd(C_n+1,C_n-1)>1$. Here the starting terms for $n \leq 4200:$ 1, 3, 7, 15, 31, 63, 127, 255, 511, 1023, 2047, 4095. Is $b_r=2^n-1?$.

Answer 1

Yes. $$\gcd(C_n+1,C_n-1)=\gcd(C_{n}+1 , 2)>1$$ So $C_{n} \equiv 1 \pmod {2}$. So you are basically looking for odd Catlan numbers.

However, it was proved here that the only $n$ for which $C_{n} \equiv 1 \pmod {2}$ is of the form $2^{m}-1$ where $m$ is an integer. So $b_{r}$ is of the form $2^{n}-1$.