



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
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OFFSET

1,2


COMMENTS

From Robert Israel, Feb 03 2016: (Start)
a(n) is the first member of the nth sequence in the greedy partition of the nonnegative integers into sequences that contain no 3term arithmetic progression.
As a special case (proved by Roth in 1953) of Szemerédi's theorem, sequences with no 3term 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. 765772.
Tanya Khovanova and Kevin Wu, Base 3/2 and Greedily Partitioned Sequences, arXiv:2007.09705 [math.NT], 2020.
K. F. Roth, On certain sets of integers, Journal of the London Mathematical Society s128 (1953), 104109.
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*xy, 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: A132605 A088157 A352017 * A079034 A265500 A212338
Adjacent sequences: A265313 A265314 A265315 * A265317 A265318 A265319


KEYWORD

nonn


AUTHOR

Max Barrentine, Dec 06 2015


STATUS

approved



