Is 3 ever a divisor or 2^n?

Note: In an effort to avoid text formatting issues, I’m using two carets to denote exponentiation. So, x^^n means x to the nth power. I hope this works.

2^^n/3. Is it ever an integer?
Intuitively, it seems not. Is it really simple to (dis)prove it and I’m thinking too hard? I tried algebra/logarithm stuff, and had no luck.

The next step I took was to make some sort of function mapping the powers of 2 to the distance between the powers of 2. That is, I wanted to show that the distance between the powers is a quantity that is a power of 2. (Ex: the distance between 2^^6 and 2^^5 is 32, which equals 2^^5. This means I’m mapping the input power to the power after it (2^^n maps to a function of 2^^(n+1) [f(2^^n)=2^^(n+1)/2 to be exact]). Showing that they’re tied to each other in this way, I thought it could prove that these numbers will never be in 3ℤ by showing the mapping goes on forever. Not sure this will get me there, though.

I was thinking of further generalizing it to any co-prime pair of integers. (gcd(n,m)=1 ⇒ m∤n^^p). Actually, you can go further. gcd(14,35) is 7, and no power of either will ever (seemingly) divide the other. 35∤14^^p & 14∤35^^p.