Solution to simple mathematical game Consider the following game (that I made up). Two players each attempt to name a target number. The first player begins by naming 1. On each subsequent turn, a player can name any larger number that differs from the current number by a divisor of the current number, including itself. For example, if the current number were 6, the next number named could be any one of 7, 8, 9, or 12. A player that names the target number wins. A player forced to name a number larger than the target number loses.
If the target number is odd, the first player can win trivially by always naming an odd number, because the difference between the current number and the target number is even, and no odd number has an even divisor. The thing I can't seem to figure out is whether the first player also has a winning strategy for every even number greater than 6, and what that strategy is if so. Obviously, one can determine the optimum course of play for any target number by working backward, and doing so seems to suggest the first player can always win, but I haven't been able to understand why.
Is there any way to prove that the first player can always win for any number greater than 6?
 A: Not an answer.
For those who want to see the nimbers of these games, I have written the following code, which calculates the nimbers for the games with target $N$ from $2$ to $999$.
How to use it:
go to this sagecell page and paste the following text there, then press "Evaluate".
def Nim(N):
    a = [0] * (N + 1)
    maxNim = 4
    for i in range(N - 1, 0, -1):
        f = [False] * maxNim
        for d in divisors(i):
            if i + d > N: continue
            f[a[i + d]] = True
        j = 0
        while j < maxNim:
            if f[j] == False: break
            j += 1
        f = []
        a[i] = j
        if j == maxNim: maxNim *= 2
    return a

for n in range(2, 1000):
    print(n, Nim(n)[1:])

Each line of the output looks like this:
(4, [0, 2, 1, 0])

This means: if the target number is $4$, then a player facing number $1, 2, 3, 4$ has nimber $0, 2, 1, 0$, respectively.

It seems that it's not clear whether my comment above about the convergence of $\{n(N, M)\}_N$ is correct.
Up to $N = 999$:
the sequence $\{n(N, 2)\}$ seems to converge after $N = 72$;
the sequence $\{n(N, 4)\}$ seems to converge after $N = 282$;
the sequence $\{n(N, 6)\}$ seems to converge after $N = 160$.
However:
the sequence $\{n(N, 8)\}$ only converges after $N = 910$.
So it's fair to say that it's still not clear whether $\{n(N, 8)\}$ converges.
Similar things happen with $\{n(N, 16)\}$ etc.
