A262057 -id:A262057 - cp's OEIS frontend

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

Max Barrentine, Dec 06 2015

nonn

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

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, and Kevin Zhao, Variants of Base 3 over 2, arXiv:1901.09818 [math.NT], 2019.
J. Gerver, J. Propp, and 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.

Cf. A006997, A262057.

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

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

A348636 Greedy Cantor's Dust Partition.

Original entry on oeis.org

1, 3, 8, 22, 24, 65, 70, 72, 194, 208, 210, 215, 580, 582, 623, 628, 630, 644, 1738, 1740, 1745, 1867, 1869, 1883, 1888, 1890, 1931, 5212, 5214, 5219, 5233, 5235, 5600, 5605, 5607, 5648, 5662, 5664, 5669, 5791, 5793, 15635, 15640, 15642, 15656, 15697, 15699

Offset: 1

Gordon Hamilton, Oct 28 2021

nonn

Starting at 1, consecutively partition the positive integers into sets s(1), s(2), s(3), ... so that no arithmetic sequence of length 3 exists in a set. When choosing s(k), always choose k as small as possible. a(n) = smallest number in s(n).

			S(1) = Cantor's dust 1,2,4,5,10,11,13,14,28,29,31,32,... (A003278)
S(2) = 3,6,7,12,15,16,19,30,33,34,...
S(3) = 8,9,17,18,20,21,35,36,44,...
S(4) = 22,23,25,26,49,50,52,53,...
S(5) = 24,27,51,54,60,63,64,67,...
S(6) = 65,66,68,69,...
S(7) = 70,71,...
S(8) = 72,...
a(1) = min [S(1)] = 1
a(2) = min [S(2)] = 3
a(3) = min [S(3)] = 8
a(4) = min [S(4)] = 22
a(5) = min [S(5)] = 24
a(6) = min [S(6)] = 65
a(7) = min [S(7)] = 70
a(8) = min [S(8)] = 72

MathPickle, Hare vs. Hare, 2017.

Cf. A003278, A005836.

One more than A265316, which is the first row of A262057.

a(n) = A265316(n) + 1.

More terms from David A. Corneth, Nov 03 2021 (computed from A265316).

cp's OEIS Frontend

A265316 First row of A262057.

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