What is known about the prime factorization of numbers of the form $2^k+1$?

Question

Let $n=2^k+1$, $k$ a positive integer. What is known about the prime factorization of these numbers? For example, consider $J(n)=\Omega(n)-\omega(n)$, where $\Omega(n)$ is the sum of multiplicities in the prime factorization of $n$, and $\omega(n)$ is the number of distinct prime factors of $n$. It would seem that $J(2^{3^i}+1)=i$ for $i$ a positive integer. Is $J(2^k+1)<J(2^{3^i})$ when $k<3^i$? Unfortunately, for these numbers we do not have Fermat's little theorem and the resulting arsenal of theorems and propositions that can be derived from it. Hence my question, is there anything we know about the prime factorization of these numbers that might help settle questions like the one above?