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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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0
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
COMMENTS
The game is equivalent to the game of Nim with the additional allowed move consisting of removing one object from each pile.
MATHEMATICA
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]
CROSSREFS
Cf. A003557, A227691, A227763, A227764, A171947, A005240, A081691, A099352, A171945, A171949, A285304, A120442, A137295, A275432.
Exceptional N-positions for Epstein's Put or Take a Square game.
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0
11, 44, 52, 71, 84, 92, 136, 208, 252, 284, 291, 296, 436, 444, 468, 491, 601, 704, 832, 918, 933, 976, 1164, 1169, 1184, 1276, 1291, 1558, 1684, 1699, 1708, 1724, 1837, 1856, 2028, 2080, 2123, 2389, 2412, 2536, 2619, 2624, 2664
REFERENCES
R. K. Guy, Letter to N. J. A. Sloane, Aug 01, 1975.
Remoteness number of n in Simon Norton's game of Tribulations.
(Formerly M0178)
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3
0, 1, 2, 1, 6, 3, 1, 5, 3, 2, 1, 2, 3, 4, 3, 1, 9, 3, 6, 7, 8, 1, 10, 3, 2, 3, 4, 5, 1, 4, 3, 8, 7, 5, 9, 7, 1, 14, 3, 4, 7, 4, 2, 9, 4, 1, 2, 3, 4, 7, 8, 12, 16, 9, 3, 1, 12, 3, 14, 7, 6, 4, 8, 6, 3, 2, 1, 6, 3, 5, 7, 11, 4
COMMENTS
The game of Tribulations is similar to Epstein's game in A005240, but the number of chips to be put or taken is the largest triangular number not larger than C: C-> C +- A057944(C). The remoteness is the number of moves in the game if the initial heap has n chips and both players play the optimum strategy. - R. J. Mathar, May 06 2016
REFERENCES
E. R. Berlekamp, J. H. Conway and R. K. Guy, Winning Ways, Academic Press, NY, 2 vols., 1982, see p. 502.
R. K. Guy, Fair Game: How to play impartial combinatorial games, COMAP's Mathematical Exploration Series, 1989; see p. 88.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
EXAMPLE
For all positive triangular numbers ( A000217) the remoteness is 1, because the starting player uses the strategy to take all of the chips and the game is over. The remoteness of 2 is 2, because taking one or putting one in the first move leads anyway to a n with remoteness 1. The remoteness of 4 is 6: 4 -> 7 -> 13 -> 23 -> 2 -> (1 or 3) -> 0. - R. J. Mathar, May 06 2016
CROSSREFS
See A266726 for indices of even-valued terms (losing positions).
N-positions in Epstein's Put or Take a Square game.
(Formerly M3350)
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2
1, 4, 9, 11, 14, 16, 21, 25, 30, 36, 41, 44, 49, 52, 54, 64, 69, 71, 81, 84, 86, 92, 100, 105, 120, 121, 126, 136, 141, 144, 149, 164, 169, 174, 189, 196, 201, 208, 216, 225, 230, 245, 252
REFERENCES
R. K. Guy, Unsolved Problems in Number Theory, E26.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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