$n$ divides $2^{2^n+1}+1$ $\implies n$ divides $2^{2^{2^n+1}+1}+1$?

There are two ways to try to prove this. One is above, the other is its de Morgan counterpart: $n$ doesn't divide $2^{2^{2^n+1}+1}+1 \implies$ $n$ doesn't divide $2^{2^n+1}+1$. Disproving it requires only one example of course.

Tried using $\gcd(2^a+1, 2^b+1) = 2^{\gcd(a, b)}+1$ (where $a$ and $b$ are odd positive integers), stuck on both ends. I figured out that if $n$ divides $2^n+1$ then n divides both $2^{2^n+1}+1$ and $2^{2^{2^n+1}+1}+1$ but this implication doesn't work backwards (e.g. $n=57$).

Would appreciate some help.