This is a generalized version of a TED-ED puzzle.

You just wrote a very good puzzle on PSE, getting three upvotes a week. However, the puzzle noob hates your puzzles, and sends you to their dimension of “bad” puzzles.

When you reach the dimension, the noob lays down $10 \ge n \ge 100$ boxes, with $1$ through $n$ coins on the ground. Each round, you take one of the boxes, and the noob takes every box which divides it. For example, if you take the box with $12$ coins, boxes $1$, $2$, $3$, $4$ and $6$ all go to the noob. You can only take boxes with at least one other untaken box dividing it. For example, if the only boxes left are $3$, $9$ and $97$, you cannot take the $97$.

When you run out of moves, the noob takes every untaken box. And if your boxes have more coins altogether than the noob’s, you win and get to go home, then write and solve more puzzles. If not, you will lose stuck in the dimension forever, which apparently has no access to PSE.

Question: for which values of $n$ do you have a winning strategy, and what are the strategies for varying values of $n$ (partial answers welcome)?