In the interest of bringing the question to a conclusion, let me say that it is an immediate consequence of Easton's theorem, as mentioned in the comments, that the various GCH assertions at the $\aleph_n$ or indeed at any successor cardinals are independent of each other—any GCH pattern whatsoever on successor cardinals can be achieved by forcing.

For example, you can have the GCH hold at $\aleph_n$ exactly when $n$ is prime, or a perfect square, or prime power, or any pattern at all. They are independent.

In the interest of bringing the question to a conclusion, let me say that it is an immediate consequence of Easton's theorem, as mentioned in the comments, that the various GCH assertions at the $\aleph_n$ or indeed at regular cardinals generally are independent of each other—any GCH pattern whatsoever on regular cardinals can be achieved by forcing.

For example, you can have the GCH hold at $\aleph_n$ exactly when $n$ is prime, or a perfect square, or prime power, or any pattern at all. They are independent.