A005240 -id:A005240

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A006019

Remoteness number of n in Simon Norton's game of Tribulations.
(Formerly M0178)

+10
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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

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).

LINKS

R. J. Mathar, Table of n, a(n) for n = 0..9999

R. J. Mathar, JAVA Program calculating A006019

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).

KEYWORD

nonn

AUTHOR

N. J. A. Sloane

EXTENSIONS

Name and offset corrected by N. J. A. Sloane, Jan 03 2016

STATUS

approved

A005241

N-positions in Epstein's Put or Take a Square game.
(Formerly M3350)

+10
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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

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).

LINKS

Table of n, a(n) for n=1..43.

R. K. Guy, Letter to N. J. A. Sloane, Aug 1975

CROSSREFS

Cf. A005240.

KEYWORD

nonn

AUTHOR

N. J. A. Sloane.

STATUS

approved

A259963

Exceptional N-positions for Epstein's Put or Take a Square game.

+10
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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

REFERENCES

R. K. Guy, Letter to N. J. A. Sloane, Aug 01, 1975.

LINKS

Table of n, a(n) for n=1..43.

R. K. Guy, Letter to N. J. A. Sloane, Aug 1975

CROSSREFS

A subset of A005241. Cf. A005242.

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Jul 10 2015

STATUS

approved

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.

+10
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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

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

LINKS

Table of n, a(n) for n=1..57.

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.

KEYWORD

nonn

AUTHOR

José María Grau Ribas, Jan 21 2021

STATUS

approved

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