A227691 Winning positions in the Subtract-a-Prime game.

2, 3, 4, 5, 6, 7, 8, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 26, 27, 28, 29, 30, 31, 32, 33, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 50, 51, 52, 53, 54, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74

Offset: 1

Author

José María Grau Ribas, Jul 19 2013

Keywords

• nonn

Comments

Consider the following game: two players make moves in turn, initially the number on the board is n. Each move consists of subtracting a prime number that is at most the number on the board. The player who cannot play loses. This sequence is the set of winner positions in this game.

Complement of A025043.

Crossrefs

The Grundy numbers of this game are in A014589.

Programs

Mathematica

moves[n_] := Table[n - Prime[i], {i, 1, PrimePi[n]}]
gana[n_] := gana[n] = If[n < 2, False,! Select[moves[n],!gana[#] &] == {}];
Select[Range[155], gana[#] &]

PARI

is(n)=if(isprime(n) || isprime(n-1), return(1)); if(n<15,return(0)); for(k=9,n-1,if(isprime(n-k) && !is(k), return(1))); 0 \\ Charles R Greathouse IV, Nov 13 2013