

A030193


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


3



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
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OFFSET

0,2


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 FurstenbergSárközy 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
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 "takeaway", J. Combinatorial Theory, 1 (1966), 443458.
I. Ruzsa, Difference sets without squares, Period. Math. Hungar. 15 (1984), no. 3, 205209.
A. Sárközy, On difference sets of sequences of integers I, Acta Mathematica Academiae Scientiarum Hungarica, March 1978, Volume 31, Issue 1, pp 125149.
A. Sárközy, On difference sets of sequences of integers III, Acta Mathematica Academiae Scientiarum Hungarica, September 1978, Volume 31, Issue 3, pp 355386.
A. Sárközy, On the difference sets of sequences of integers II, Eotvos Sect. Math. 21(1978), 4553.


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 *)


CROSSREFS

Sequence in context: A038126 A047215 A059536 * A292653 A028250 A190087
Adjacent sequences: A030190 A030191 A030192 * A030194 A030195 A030196


KEYWORD

nonn


AUTHOR

Jan Kristian Haugland (jankrihau(AT)hotmail.com)


EXTENSIONS

More terms from Karl W. Heuer, Jun 13 2013


STATUS

approved



