I know that there are no solutions to $2^n\equiv 1\pmod{n}$ for $n>1$ and I can prove that there are infinitely many $n$ such that $2^{n+1}\equiv1\pmod{n}$.
My question is:

Do we know other fixed values $k\in \mathbb{N}$ such that $2^{n+k}\equiv 1\pmod{n}$ holds infinitely often?