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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 Szemer├ędi'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, Szemer├ędi'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 April 24 22:26 EDT 2019. Contains 322446 sequences. (Running on oeis4.)