Here is a game for 2 or more players.
Start with a deck of cards with n cards labeled 1 to n.
On each move the player whose move it is draws one card.
Do not replace card.
Write down first number drawn.
On each move (after the first) a player draws a card to get
the value k.
He/she may either add k to the written down number,
subtract k from the written down number (as long as the
difference is not negative), multiply k with the written down
number, or divide the written down number by k
(as long as k divides evenly into the written down number).
A new written down number, in this way, is obtained.
A player gets a point every time the written down number
they generate is a prime number.
Play continues for a total of n moves (when the deck
of cards runs out).
Here is an example game (for 2 players). (n=10)
First move: Player 1 draws 7. (Gets a point)
2nd move: Player 2 draws a 4. 7 + 4 = 11. (Gets a point)
3rd move:Player 1 draws a 1. 11/1 = 11. (Gets a point)
4th move: Player 2 draws a 9. 11-9 = 2. (Gets a point)
5th move: Player 1 draws a 2. 2/2=1. (No point)
6th move: Player 2 draws a 3. 1*3 = 3. (Gets a point)
7th move: Player 1 draws a 10. 3+10=13. (Gets a point)
8th move: Player 2 draws a 5. 13*5= 65. (No point)
9th move: Player 1 draws a 8. 65+8=73. (Gets a point)
10th move: Player 2 draws a 6. 73-6=67. (Gets a point)
So player 1 gets 4 points, player 2 gets 4 points too, a tie.
(What do you expect when I play myself?)
In practice, it would probably be better to have a higher n.
Also, if you prefer a pure-strategy game without luck, you can,
instead of using cards, have the integers "drawn" be from a
predetermined sequence (such as 1,2,3,4,....,n or such as
the primes taken in order.)