When is the "Taxman Game" winnable? I recently came across the "Taxman Game" the rules are in the link, but I'll repeat them here:

We start with a pile of integers, from 1 to some number that you
  choose [$n$]. You take one, and I get all the others that divide it evenly.
  We repeat until there's nothing left. BUT (big but!) I have to get
  something on each turn. So when none of the numbers in the list have
  any divisors that have not been taken, I take them all. Your score is
  the sum of all the numbers you took, while my score is the sum of all
  the numbers I took.

The 'you' and 'I' is a bit confusing, there is only one player here - the 'you' - as the 'taxman' ('I') side is entirely deterministic.
The question I have is this: for what $n$ is the game winnable? I think that it is winnable for all $n>3$ but I'm not sure how to go about proving this. At $n=3$, the best strategy is a draw (pick 3, taxman get 1 and 2), while for $n=1$, the player has no winning move so the lowest bound on winnable $n$ must be $n>3$ but is there a higher $n$ that is unwinnable?
I'd also be interested in whether there is a 'best' strategy guaranteed to give a win where it is possible and the largest possible winning margin when doing so.

Robert Moniot's "greedy heuristic" seems like a good candidate for a winning strategy. He computed the optimal strategy for all n up to 49, but he did not find a simple rule to describe it for arbitrary n.
There's an interesting discussion thread on /r/mathriddles about the taxman game. No proofs of optimality, but I thought I'd share the link since there's a number of suggested strategies on there.
