A265316 - OEIS

%I #34 Jan 06 2024 12:06:56

%S 0,2,7,21,23,64,69,71,193,207,209,214,579,581,622,627,629,643,1737,

%T 1739,1744,1866,1868,1882,1887,1889,1930,5211,5213,5218,5232,5234,

%U 5599,5604,5606,5647,5661,5663,5668,5790,5792,15634,15639,15641,15655,15696,15698

%N First row of A262057.

%C From _Robert Israel_, Feb 03 2016: (Start)

%C 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.

%C 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.

%C a(n+1) >= a(n) + 2.  There seem to be many cases where this is an equality. (End)

%C 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

%C Indices of records in A006997. - _Rémy Sigrist_, Jan 06 2024

%H Robert Israel, <a href="/A265316/b265316.txt">Table of n, a(n) for n = 1..140</a>

%H Matvey Borodin, Hannah Han, Kaylee Ji, Tanya Khovanova, Alexander Peng, David Sun, Isabel Tu, Jason Yang, William Yang, Kevin Zhang, Kevin Zhao, <a href="https://arxiv.org/abs/1901.09818">Variants of Base 3 over 2</a>, arXiv:1901.09818 [math.NT], 2019.

%H J. Gerver, J. Propp, J. Simpson, <a href="https://doi.org/10.1090/S0002-9939-1988-0929018-1">Greedily partitioning the natural numbers into sets free of arithmetic progressions</a>, Proc. of the Amer. Math. Soc. 102 (1988), no. 3, pp. 765-772.

%H Tanya Khovanova and Kevin Wu, <a href="https://arxiv.org/abs/2007.09705">Base 3/2 and Greedily Partitioned Sequences</a>, arXiv:2007.09705 [math.NT], 2020.

%H K. F. Roth, <a href="https://doi.org/10.1112/jlms/s1-28.1.104">On certain sets of integers</a>, Journal of the London Mathematical Society s1-28 (1953), 104-109.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Szemer%C3%A9di%27s_theorem"> Szemerédi's theorem</a>.

%F A006997(a(n)) = n - 1. - _Rémy Sigrist_, Jan 06 2024

%p M:= 100: # to get a(1) to a(M)

%p for i from 1 to M do B[i]:= {}: F[i]:= {}: od:

%p for x from 0 do

%p   for i from 1 to M do

%p      if not member(x,F[i]) then

%p        F[i]:= F[i] union map(y -> 2*x-y, B[i]);

%p      B[i]:= B[i] union {x};

%p      if not assigned(A[i]) then A[i]:= x fi;

%p      break

%p     fi

%p   od;

%p   if i = M+1 then break fi;

%p od:

%p seq(A[i],i=1..M); # _Robert Israel_, Feb 03 2016

%Y Cf. A006997, A262057.

%K nonn

%O 1,2

%A _Max Barrentine_, Dec 06 2015