A030193 Let S = squares; a(0)=0; a(n) = smallest m such that m - a(i) is not in S for any i < n.

0, 2, 5, 7, 10, 12, 15, 17, 20, 22, 34, 39, 44, 52, 57, 62, 65, 67, 72, 85, 95, 109, 119, 124, 127, 130, 132, 137, 142, 147, 150, 170, 177, 180, 182, 187, 192, 197, 204, 210, 215, 238, 243, 249, 255, 257, 260, 262, 267

Offset: 0

Author

Jan Kristian Haugland

Keywords

• nonn

Comments

Consider the following game: two players make moves in turn, initially the number on the board is n, each move consists of subtracting a perfect square from the number on the board, the player who faces 0 loses. This sequence is the set of losing positions in this game. - Mikhail Dvorkin (mikhail.dvorkin(AT)gmail.com), Jan 27 2008

This sequence was investigated by Golomb (1966), who proved that it is infinite. More strongly (as Ruzsa 1984 notes) the number of values up to any given n is at least proportional to sqrt(n). No two numbers in this sequence differ by a square, and this sequence can be defined as the lexicographically first (greedy) sequence with no square differences. It follows from the Furstenberg-Sárközy theorem (e.g., see Sárközy 1978) that its natural density is zero. - David Eppstein, Nov 20 2016

Links

• Karl W. Heuer, Table of n, a(n) for n = 0..61299
• Code Golf Stack Exchange, First sequence with no square differences, 2021.
• Ben Green and Mehtaab Sawhney, Improved bounds for the Furstenberg-Sárközy theorem, arXiv:2411.17448 [math.NT], 2024. See p. 2.
• David Eppstein, Faster Evaluation of Subtraction Games, Proceedings of the 9th International Conference on Fun with Algorithms (FUN 2018), Leibniz International Proceedings in Informatics, arXiv:1804.06515 [cs.DS], 2018.
• S. W. Golomb, A mathematical investigation of games of "take-away", J. Combinatorial Theory, 1 (1966), 443-458.
• I. Ruzsa, Difference sets without squares, Period. Math. Hungar. 15 (1984), no. 3, 205-209.
• A. Sárközy, On difference sets of sequences of integers I, Acta Mathematica Academiae Scientiarum Hungarica, March 1978, Volume 31, Issue 1, pp 125-149.
• A. Sárközy, On difference sets of sequences of integers III, Acta Mathematica Academiae Scientiarum Hungarica, September 1978, Volume 31, Issue 3, pp 355-386.
• A. Sárközy, On the difference sets of sequences of integers II, Eotvos Sect. Math. 21(1978), 45-53.

Crossrefs

Cf. A000037, A000290.

Programs

Mathematica

moves[n_] := Table[n - i^2, {i, 1, Sqrt[n]}]; gana[n_] := Which[n == 0, False, True,!Select[moves[n], !gana[#] &] =={}]; Select[Range[155] - 1, ! gana[#] &] (* José María Grau Ribas, Jul 19 2013 *)
Nest[Append[#, Block[{k = Last[#]}, While[AnyTrue[k - #, IntegerQ@ Sqrt@ # &], k++]; k]] &, {0}, 48] (* Michael De Vlieger, Jul 11 2018 *)

Extensions

More terms from Karl W. Heuer, Jun 13 2013