Here is a game for two players.
The players take turns picking any integer from 1,2,3,...., r that has not been picked (by either player) previously in the game, where r is some large integer (such as 1000 or 10000 or more).
Let this picked number be m. (m was picked by the player temporarily called the "provider".)
The same player then picks any integer k where 1 <= k <= m. (k may have been picked earlier any number of times in the game.)
The other player (the "finder") then tries to find any positive integer n not equal to m (n can be arbitrarily large and either picked previously during the game or not) such that:
Both d(m) = d(n) and d(m+k) = d(n+k), where d(j) is the number of divisors of j.
The finder may also provide a proof that no such n (not equal to m) exists.
If the finder either finds an n or proves there is no such n fitting the conditions, then the finder gets a point.
Players then switch who is the provider and who is the finder.
Players play an even predetermined number of moves, and the player with the largest score wins.