It is sufficient to find a pair of twin primes $p, q$ such that the orders of $2$ modulo $p$ and $q$ are odd, as then we can take $n$ to be the product of the two orders. Notice that it is a necessary condition that $2^{\frac{p-1}{2}}\equiv 1 \pmod p$ and $2^{\frac{q-1}{2}}\equiv 1 \pmod q$, i.e. that $2$ is a quadratic residue both modulo $p, q$, so $p, q \equiv 1, 7 \pmod {8}$. By hand we check that the smallest such pair is $71, 73$. It works because $\nu_2(70)=1$, so the order of $2$ modulo $71$ divides $35$ by our construction, and the order of $2$ modulo $73$ is also odd since $73 \mid 2^{9}-1$.