A265316 First row of A262057.

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

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

Indices of records in A006997. - Rémy Sigrist, Jan 06 2024

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.

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 s1-28 (1953), 104-109.

Wikipedia, Szemerédi's theorem.

Formula

A006997(a(n)) = n - 1. - Rémy Sigrist, Jan 06 2024

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. A006997, 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