A285304 -id:A285304 - cp's OEIS frontend

A285847 N-positions in the sum-from-product game.

6, 8, 9, 10, 12, 14, 15, 18, 20, 21, 24, 26, 28, 30, 33, 35, 36, 38, 39, 40, 42, 44, 45, 48, 50, 51, 54, 56, 58, 60, 62, 63, 65, 66, 68, 69, 70, 72, 75, 76, 77, 78, 80, 84, 86, 87, 90, 91, 93, 95, 96, 98, 99, 100, 102, 104, 105, 108, 110, 111

Offset: 1

Tanya Khovanova and students, May 06 2017

nonn

The sum-from-product game is played by two players alternating moves. Given a positive integer n, a player can choose any two integers a and b, such that ab=n. The player subtracts a+b from n, given that the result is positive. That is, the next player starts with a new number n-a-b. A player without a move loses.
Prime numbers are P-positions.
P-positions are A285304.

			Numbers 1, 2, 3, 4, 5, 7, 11 are P-positions as there are no legal moves. Therefore, 6 and 8 are N-positions, as the only move from 6 goes to 1, and the only move from 8 goes to 2. It follows that 16 is a P-position as there are two moves: 16-4-4 = 8, and 16-2-8 = 6: both are N-positions.

Pratik Alladi, Neel Bhalla, Tanya Khovanova, Nathan Sheffield, Eddie Song, William Sun, Andrew The, Alan Wang, Naor Wiesel, Kevin Zhang Kevin Zhao, PRIMES STEP Plays Games, arXiv:1707.07201 [math.CO], 2017, Section 6.

Cf. A285304.

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]

cp's OEIS Frontend

A285847 N-positions in the sum-from-product game.

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