A227691 -id:A227691 - cp's OEIS frontend

A227455 Sequence defined recursively: 1 is in the sequence, and k > 1 is in the sequence iff for some prime divisor p of k, p-1 is not in the sequence.

1, 3, 5, 6, 9, 10, 12, 15, 17, 18, 20, 21, 23, 24, 25, 27, 29, 30, 33, 34, 35, 36, 39, 40, 42, 45, 46, 48, 50, 51, 53, 54, 55, 57, 58, 60, 63, 65, 66, 68, 69, 70, 72, 75, 78, 80, 81, 83, 84, 85, 87, 89, 90, 92, 93, 95, 96, 99, 100, 102, 105, 106, 108, 110

Offset: 1

José María Grau Ribas, Jul 12 2013

nonn

Consider a two-player game in which players take turns and a player given the position k = p_1^s_1 * ... * p_j^s_j must choose one of the j possible moves p_1 - 1, ..., p_j - 1, and the player's chosen move becomes the position given to the other player. The first player whose only possible move is 1 loses. Terms in this sequence are the winning positions for the player whose turn it is.

			Numbers of the form 2^k are not in the sequence because their unique prime divisor is p = 2 and p-1 = 1 is in the sequence.
Numbers of the form 3^k are in the sequence because 3-1 = 2 is not in the sequence.
Numbers of the form 5^k are in the sequence because 5-1 = 4 = 2^2, and 2 is not in the sequence.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A227006, A227691, A027748.

a227455 n = a227455_list !! (n-1)
a227455_list = 1 : f [2..] [1] where
   f (v:vs) ws = if any (`notElem` ws) $ map (subtract 1) $ a027748_row v
                    then v : f vs (v : ws) else f vs ws
-- Reinhard Zumkeller, Dec 08 2014

fa=FactorInteger;win[1] = True; win[n_] := win[n] = ! Union@Table[win[fa[n][[i, 1]] - 1], {i, 1, Length@fa@n}] == {True}; Select[Range[300], win]

Edited by Jon E. Schoenfield, Jan 23 2021

A227763 Winning positions in the misere version of the Subtract-a-Prime game.

Original entry on oeis.org

1, 4, 5, 6, 7, 8, 9, 10, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 28, 29, 30, 31, 32, 33, 34, 35, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 52, 53, 54, 55, 56, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75

Offset: 1

José María Grau Ribas, Jul 29 2013

nonn

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 wins. This sequence is the set of winning positions in this game.

Cf. A227691, A025043, A227764.

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

A227764 Losing positions in the misere version of the Subtract-a-Prime game.

Original entry on oeis.org

2, 3, 11, 12, 27, 36, 37, 51, 57, 87, 93, 102, 117, 123, 127, 135, 147, 157, 171, 177, 189, 197, 207, 219, 237, 249, 255, 261, 267, 291, 297, 303, 311, 312, 321, 327, 337, 345, 357, 363, 377, 387, 393, 397, 405, 417, 427, 447, 453, 471, 477, 483, 487, 495

Offset: 1

José María Grau Ribas, Jul 29 2013

nonn

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 wins. This sequence is the set of lost positions in this game.

Cf. A227691, A025043, A227763.

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

A340780 Losing positions n (P-positions) in the following game: two players take turns dividing the current value of n by either a prime power > 1 or by A007947(n) to obtain the new value of n. The winner is the player whose division results in 1.

Original entry on oeis.org

1, 12, 18, 20, 28, 44, 45, 50, 52, 63, 68, 75, 76, 92, 98, 99, 116, 117, 120, 124, 147, 148, 153, 164, 168, 171, 172, 175, 188, 207, 212, 216, 236, 242, 244, 245, 261, 264, 268, 270, 275, 279, 280, 284, 292, 312, 316, 325, 332, 333, 338, 356, 363, 369, 378, 387, 388

Offset: 1

José María Grau Ribas, Jan 21 2021

nonn

The game is equivalent to the game of Nim with the additional allowed move consisting of removing one object from each pile.

Cf. A003557, A227691, A227763, A227764, A171947, A005240, A081691, A099352, A171945, A171949, A285304, A120442, A137295, A275432.

Clear[moves,los]; A003557[n_]:= {Module[{aux = FactorInteger[n], L=Length[FactorInteger[n]]},Product[aux[[i,1]]^(aux[[i, 2]]-1),{i, L}]]};
moves[n_] :=moves[n] = Module[{aux = FactorInteger[n], L=Length[ FactorInteger [n]]}, Union[Flatten[Table[n/aux[[i,1]]^j, {i,1,L},{j,1,aux[[i,2]]}],1], A003557[n]]]; los[1]=True; los[m_] := los[m] = If[PrimeQ[m], False, Union@Flatten@Table[los[moves[m][[i]]], {i,1,Length[moves[m]]}] == {False}]; Select[Range[400], los]

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A227455 Sequence defined recursively: 1 is in the sequence, and k > 1 is in the sequence iff for some prime divisor p of k, p-1 is not in the sequence.

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A227763 Winning positions in the misere version of the Subtract-a-Prime game.

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A227764 Losing positions in the misere version of the Subtract-a-Prime game.

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A340780 Losing positions n (P-positions) in the following game: two players take turns dividing the current value of n by either a prime power > 1 or by A007947(n) to obtain the new value of n. The winner is the player whose division results in 1.

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