Find all relatively prime successive binomial coefficients

Question

Find all couples $(n,k)$ , $n \geq 2$ and $0\leq $ k $ \leq n - 1$, such as :
$\binom nk$ and $\binom {n}{k+1}$ are relatively prime.
I first tried finding a link between the two using combinatorics, but it wasn't that easy, since they don't have a particular order, and any could be greater than the other.
Then I recalled that if $n$ is prime, it's guaranteed that it divides both of them, I tried to apply this in a more general scenario, using a prime divisor of $n$, but it didn't succeed.
Any ideas or tips?