A275432 -id:A275432

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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.

+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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