A099352 -id:A099352 - cp's OEIS frontend

A099353 From P-positions in a certain game.

0, 2, 7, 12, 18, 25, 35, 45, 56, 68, 83, 98, 114, 131, 149, 170, 191, 213, 236, 260, 285, 313, 341, 370, 400, 431, 463, 496, 530, 565, 603, 641, 680, 720, 761, 803, 846, 890, 935, 983, 1031, 1080, 1130, 1181, 1233, 1286, 1340, 1395, 1451, 1510, 1569

Offset: 0

N. J. A. Sloane, Nov 16 2004

Nathaniel Johnston, Table of n, a(n) for n = 0..10000
A. S. Fraenkel, New games related to old and new sequences, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 4, Paper G6, 2004.

a:=proc(n) option remember: local j,t: if(n=0)then return 0: else t:=a(n-1)+1: for j from 0 to n-1 do if(t=b(j))then return t+1: elif(tNathaniel Johnston, Apr 28 2011

a[n_] := a[n] = Module[{j, t}, If[n == 0, 0, t = a[n - 1] + 1; For[j = 0, j <= n - 1, j++, Which[t == b[j], Return[t + 1], t < b[j], Break[]]]; t]];
b[n_] := b[n] = If[n == 0,  0, b[n - 1] + a[n] - Floor[(a[n - 1] + 1)/a[n]] + 2];
Table[b[n], {n, 0, 50}] (* Jean-François Alcover, Mar 10 2023, after Nathaniel Johnston *)

See A099352.

A099356 From P-positions in a certain game.

Original entry on oeis.org

0, 1, 3, 4, 6, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 52, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69

Offset: 0

N. J. A. Sloane, Nov 16 2004

Nathaniel Johnston, Table of n, a(n) for n = 0..10000
A. S. Fraenkel, New games related to old and new sequences, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 4, Paper G6, 2004.

Cf. A099352 - A099355.

a:=proc(n) option remember: local j, t: if(n=0)then return 0: else t:=a(n-1)+1: for j from 0 to n-1 do if(t=b(j))then return t+1: elif(tNathaniel Johnston, Apr 28 2011

Let a(n) = this sequence, b(n) = A099357. Then a(n) = the smallest number not in {a(0), b(0), a(1), b(1), ..., a(n-1), b(n-1)}; b(n) = b(n-1) + (-1)^a(n-1)*a(n-1) + a(n) + 1. Apart from initial zero, complement of A099357.

A099357 From P-positions in a certain game.

Original entry on oeis.org

0, 2, 5, 7, 18, 33, 51, 53, 75, 77, 103, 105, 135, 137, 171, 174, 176, 218, 220, 266, 268, 318, 320, 374, 376, 434, 436, 498, 500, 567, 637, 639, 713, 715, 793, 795, 877, 879, 965, 967, 1057, 1059, 1153, 1155, 1253, 1255, 1358, 1465, 1575, 1577

Offset: 0

N. J. A. Sloane, Nov 16 2004

Table 10 in Fraenkel is incorrect from n=6 onward. - Nathaniel Johnston, Apr 28 2011

Nathaniel Johnston, Table of n, a(n) for n = 0..10000
A. S. Fraenkel, New games related to old and new sequences, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 4, Paper G6, 2004.

Cf. A099352 - A099356.

a:=proc(n) option remember: local j, t: if(n=0)then return 0: else t:=a(n-1)+1: for j from 0 to n-1 do if(t=b(j))then return t+1: elif(tNathaniel Johnston, Apr 28 2011

See A099356.

Edited and corrected by Nathaniel Johnston, Apr 28 2011

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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A099353 From P-positions in a certain game.

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A099356 From P-positions in a certain game.

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A099357 From P-positions in a certain 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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