Here is a game for any number of players. (I wonder, though, if having an even number of players gives an advantage to some player(s).)
Needed: A deck of n flash cards labeled 1, 2, 3, ..., n, one number per card, where n is a multiple of the number of players.
The cards are placed face-up in order in a row between the players.
Players take turns, switching one pair of cards each move.
On the m-th move of the game, a player switches the positions the card labeled with the number m and any other card.
The player moving gets a point if both:
One of these cards he/she switched (either card; let k be the number on this card) is adjacent to a card with a number coprime to k, if the card is now at the end of the row of cards, or card k is now between two cards both with numbers coprime to k;
(In other words, card k is NOT non-coprime to any card it is now adjacent to.)
The other card switched (with the number j on it) is non-coprime to exactly one number it is now adjacent to. (Either the other number that card-j is adjacent to is coprime to j, if card j is not at the end of the row, or card j is at the end of the row.)
After n total moves, the player with the most points wins.
Clarification: m equals either k or j, not both.
Each move results in a permutation of the numbers 1 through n.
I suggest that n be large enough to make this game relatively interesting, of course.