Edit (Nov 1, 2015): Bounty awarded, but the full question (i.e., what is the optimal strategy) remains open at the time of this update.
Consider the Factor Game played as follows:
Given a list of positive integers $1, \ldots, n$, two players (red and blue) alternate turns.
On each turn, the player picks and circles (in their color) any un-circled number $k$ that has at least one proper divisor yet to be circled; their opponent responds by circling (in their own color) all remaining proper divisors of $k$.
Then they switch (colors adjusted accordingly) and the game continues until there are no legal moves remaining.
At that point, each player adds up their circled numbers and the greater sum wins.
Question: What is the optimal strategy in the Factor Game?
My guess is that, for large enough $n$, picking the number that nets the most points on each individual turn is optimal. For example, if $n = 49$, then the first player would pick $47$ (to net $46$) and their opponent would respond with $49$ (to net $42$). But this strategy is admittedly short-sighted.
(Clearly the strategy fails for some small choices of $n$ (the example of $n = 4$ is provided in the comments; $n = 9$ is another failure) but I'm okay with assuming $n$ is sufficiently large.)
I am hoping for a description of an optimal strategy and a proof that it is optimal (for $n$ large enough), but would be interested in special cases, heuristic arguments for what would be an optimal strategy, and heuristic arguments for why this problem is difficult.
Motivation: This game is sometimes played in elementary schools (or, in my classes, with elementary school teachers) to explore concepts like prime, composite, divisor, proper divisor, etc. In such a case, the numbers are usually presented in an array - often a square array, and so my particular interest is when $n$ is a square.
You can play the Factor Game through the NCTM (National Council of Teachers of Mathematics) Illuminations site if you would like to experience it for yourself.
Here is a sample board when $n = 49$ (presented as a $7 \times 7$ array):