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 A265316 First row of A262057. 2
 0, 2, 7, 21, 23, 64, 69, 71, 193, 207, 209, 214, 579, 581, 622, 627, 629, 643, 1737, 1739, 1744, 1866, 1868, 1882, 1887, 1889, 1930, 5211, 5213, 5218, 5232, 5234, 5599, 5604, 5606, 5647, 5661, 5663, 5668, 5790, 5792, 15634, 15639, 15641, 15655, 15696, 15698 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS From Robert Israel, Feb 03 2016: (Start) a(n) is the first member of the n-th sequence in the greedy partition of the nonnegative integers into sequences that contain no 3-term arithmetic progression. As a special case (proved by Roth in 1953) of Szemerédi's theorem, sequences with no 3-term arithmetic progressions must have density 0.  In particular, the nonnegative integers can't be partitioned into finitely many such sequences.  Therefore this sequence is infinite. a(n+1) >= a(n) + 2.  There seem to be many cases where this is an equality. (End) It can be deduced from the main result of Gerver, Propp, Simpson (below) that a(3n+1) = 3a(2n+1), a(3n+2) = 2 + 3a(2n+1), and a(3n) = 1 + 3a(2n). This implies infinitely many cases where a(n+1) = a(n) + 2. - C. Kenneth Fan, Dec 09 2018 LINKS Robert Israel, Table of n, a(n) for n = 1..140 Matvey Borodin, Hannah Han, Kaylee Ji, Tanya Khovanova, Alexander Peng, David Sun, Isabel Tu, Jason Yang, William Yang, Kevin Zhang, Kevin Zhao, Variants of Base 3 over 2, arXiv:1901.09818 [math.NT], 2019. J. Gerver, J. Propp, J. Simpson, Greedily partitioning the natural numbers into sets free of arithmetic progressions, Proc. of the Amer. Math. Soc. 102 (1988), no. 3, pp. 765-772. K. F. Roth, On certain sets of integers, Journal of the London Mathematical Society s1-28 (1953), 104-109. Wikipedia, Szemerédi's theorem. MAPLE M:= 100: # to get a(1) to a(M) for i from 1 to M do B[i]:= {}: F[i]:= {}: od: for x from 0 do   for i from 1 to M do      if not member(x, F[i]) then        F[i]:= F[i] union map(y -> 2*x-y, B[i]);      B[i]:= B[i] union {x};      if not assigned(A[i]) then A[i]:= x fi;      break     fi   od;   if i = M+1 then break fi; od: seq(A[i], i=1..M); # Robert Israel, Feb 03 2016 CROSSREFS Cf. A262057. Sequence in context: A139012 A132605 A088157 * A079034 A265500 A212338 Adjacent sequences:  A265313 A265314 A265315 * A265317 A265318 A265319 KEYWORD nonn AUTHOR Max Barrentine, Dec 06 2015 STATUS approved

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Last modified July 4 05:54 EDT 2020. Contains 335444 sequences. (Running on oeis4.)