When are $2n$ and $3n+1$ coprime?

Linear combination.
$$2(3n+1)-3(2n)=6n+2-6n=2.$$
So $d=\divisibility(2n,3n+1)$ divides $2$, hence $d\in\{1,2\}$.

When is $d=2$? Since $2n$ is always even, $d=2$ iff $3n+1$ is even, i.e. $n$ is odd.

Conclusion. $\divisibility(2n,3n+1)=1$ if $n$ is even, and $2$ if $n$ is odd. The numbers are coprime exactly when $n$ is even.