A004793 -id:A004793 - cp's OEIS frontend

A005836 Numbers whose base-3 representation contains no 2.

0, 1, 3, 4, 9, 10, 12, 13, 27, 28, 30, 31, 36, 37, 39, 40, 81, 82, 84, 85, 90, 91, 93, 94, 108, 109, 111, 112, 117, 118, 120, 121, 243, 244, 246, 247, 252, 253, 255, 256, 270, 271, 273, 274, 279, 280, 282, 283, 324, 325, 327, 328, 333, 334, 336, 337, 351, 352

Offset: 1

N. J. A. Sloane, Jeffrey Shallit

3 does not divide binomial(2s, s) if and only if s is a member of this sequence, where binomial(2s, s) = A000984(s) are the central binomial coefficients.
This is the lexicographically earliest increasing sequence of nonnegative numbers that contains no arithmetic progression of length 3. - Robert Craigen (craigenr(AT)cc.umanitoba.ca), Jan 29 2001
In the notation of A185256 this is the Stanley Sequence S(0,1). - N. J. A. Sloane, Mar 19 2010
Complement of A074940. - Reinhard Zumkeller, Mar 23 2003
Sums of distinct powers of 3. - Ralf Stephan, Apr 27 2003
Numbers n such that central trinomial coefficient A002426(n) == 1 (mod 3). - Emeric Deutsch and Bruce E. Sagan, Dec 04 2003
A039966(a(n)+1) = 1; A104406(n) = number of terms <= n.
Subsequence of A125292; A125291(a(n)) = 1 for n>1. - Reinhard Zumkeller, Nov 26 2006
Also final value of n - 1 written in base 2 and then read in base 3 and with finally the result translated in base 10. - Philippe LALLOUET (philip.lallouet(AT)wanadoo.fr), Jun 23 2007
a(n) modulo 2 is the Thue-Morse sequence A010060. - Dennis Tseng, Jul 16 2009
Also numbers such that the balanced ternary representation is the same as the base 3 representation. - Alonso del Arte, Feb 25 2011
Fixed point of the morphism: 0 -> 01; 1 -> 34; 2 -> 67; ...; n -> (3n)(3n+1), starting from a(1) = 0. - Philippe Deléham, Oct 22 2011
It appears that this sequence lists the values of n which satisfy the condition sum(binomial(n, k)^(2*j), k = 0..n) mod 3 <> 0, for any j, with offset 0. See Maple code. - Gary Detlefs, Nov 28 2011
Also, it follows from the above comment by Philippe Lallouet that the sequence must be generated by the rules: a(1) = 0, and if m is in the sequence then so are 3*m and 3*m + 1. - L. Edson Jeffery, Nov 20 2015
Add 1 to each term and we get A003278. - N. J. A. Sloane, Dec 01 2019

			12 is a term because 12 = 110_3.
This sequence regarded as a triangle with rows of lengths 1, 1, 2, 4, 8, 16, ...:
   0
   1
   3,  4
   9, 10, 12, 13
  27, 28, 30, 31, 36, 37, 39, 40
  81, 82, 84, 85, 90, 91, 93, 94, 108, 109, 111, 112, 117, 118, 120, 121
... - _Philippe Deléham_, Jun 06 2015

Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section E10, pp. 317-323.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

David W. Wilson, Table of n, a(n) for n = 1..10000 (first 1024 terms from T. D. Noe)
J.-P. Allouche, G.-N. Han, and Jeffrey Shallit, On some conjectures of P. Barry, arXiv:2006.08909 [math.NT], 2020.
J.-P. Allouche and Jeffrey Shallit, The ring of k-regular sequences, Theoretical Computer Sci., 98 (1992), 163-197.
J.-P. Allouche and Jeffrey Shallit, The ring of k-regular sequences, Theoretical Computer Sci., 98 (1992), 163-197.
J.-P. Allouche, Jeffrey Shallit and G. Skordev, Self-generating sets, integers with missing blocks and substitutions, Discrete Math. 292 (2005) 1-15.
David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata, Congressus Numerantium, Vol. 206 (2010), 157-191. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n-1)-1) for n >= 2.]
Megumi Asada, Bruce Fang, Eva Fourakis, Sarah Manski, Nathan McNew, Steven J. Miller, Gwyneth Moreland, Ajmain Yamin, and Sindy Xin Zhang, Avoiding 3-Term Geometric Progressions in Hurwitz Quaternions, Williams College (2023).
Robert Baillie and Thomas Schmelzer, Summing Kempner's Curious (Slowly-Convergent) Series, Mathematica Notebook kempnerSums.nb, Wolfram Library Archive, 2008.
Noam Benson-Tilsen, Samuel Brock, Brandon Faunce, Monish Kumar, Noah Dokko Stein, and Joshua Zelinsky, Total Difference Labeling of Regular Infinite Graphs, arXiv:2107.11706 [math.CO], 2021.
Raghavendra Bhat, Cristian Cobeli, and Alexandru Zaharescu, Filtered rays over iterated absolute differences on layers of integers, arXiv:2309.03922 [math.NT], 2023. See page 16.
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.
Ben Chen, Richard Chen, Joshua Guo, Tanya Khovanova, Shane Lee, Neil Malur, Nastia Polina, Poonam Sahoo, Anuj Sakarda, Nathan Sheffield, and Armaan Tipirneni, On Base 3/2 and its Sequences, arXiv:1808.04304 [math.NT], 2018.
Karl Dilcher and Larry Ericksen, Hyperbinary expansions and Stern polynomials, Elec. J. Combin, Vol. 22 (2015), Article P2.24.
P. Erdős, V. Lev, G. Rauzy, C. Sandor, and A. Sarkozy, Greedy algorithm, arithmetic progressions, subset sums and divisibility, Discrete Math., Vol. 200, No. 1-3 (1999), pp. 119-135 (see Table 1). alternate link.
Joseph L. Gerver and L. Thomas Ramsey, Sets of integers with no long arithmetic progressions generated by the greedy algorithm, Math. Comp., Vol. 33, No. 148 (1979), pp. 1353-1359.
Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Identities and periodic oscillations of divide-and-conquer recurrences splitting at half, arXiv:2210.10968 [cs.DS], 2022, p. 45.
Ryota Inagaki, Tanya Khovanova, and Austin Luo, Permutation-based Strategies for Labeled Chip-Firing on k-ary Trees, arXiv:2503.09577 [math.CO], 2025. See p. 18.
Kathrin Kostorz, Robert W. Hölzel and Katharina Krischer, Distributed coupling complexity in a weakly coupled oscillatory network with associative properties, New J. Phys., Vol. 15 (2013), #083010; doi:10.1088/1367-2630/15/8/083010.
Clark Kimberling, Affinely recursive sets and orderings of languages, Discrete Math., Vol. 274, Vol. 1-3 (2004), pp. 147-160.
John W. Layman, Some Properties of a Certain Nonaveraging Sequence, J. Integer Sequences, Vol. 2 (1999), Article 99.1.3.
Manfred Madritsch and Stephan Wagner, A central limit theorem for integer partitions, Monatsh. Math., Vol. 161, No. 1 (2010), pp. 85-114.
Richard A. Moy and David Rolnick, Novel structures in Stanley sequences, Discrete Math., Vol. 339, No. 2 (2016), pp. 689-698; arXiv preprint, arXiv:1502.06013 [math.CO], 2015.
A. M. Odlyzko and R. P. Stanley, Some curious sequences constructed with the greedy algorithm, 1978, remark 1 (PDF, PS, TeX).
Paul Pollack, Analytic and Combinatorial Number Theory Course Notes, p. 228. [?Broken link]
Paul Pollack, Analytic and Combinatorial Number Theory Course Notes, p. 228.
David Rolnick and Praveen S. Venkataramana, On the growth of Stanley sequences, Discrete Math., Vol. 338, No. 11 (2015), pp. 1928-1937, see p. 1930; arXiv preprint, arXiv:1408.4710 [math.CO], 2014.
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS.
Ralf Stephan, Divide-and-conquer generating functions. I. Elementary sequences, arXiv:math/0307027 [math.CO], 2003.
Ralf Stephan, Some divide-and-conquer sequences with (relatively) simple ordinary generating functions, 2004.
Ralf Stephan, Table of generating functions.
Zoran Sunic, Tree morphisms, transducers and integer sequences, arXiv:math/0612080 [math.CO], 2006.
B. Vasic, K. Pedagani and M. Ivkovic, High-rate girth-eight low-density parity-check codes on rectangular integer lattices, IEEE Transactions on Communications, Vol. 52, Issue 8 (2004), pp. 1248-1252.
Eric Weisstein's World of Mathematics, Central Binomial Coefficient.
Index entries for 3-automatic sequences.

Cf. A039966 (characteristic function).
Cf. A002426, A004793, A005823, A007088, A007089, A032924, A033042-A033052, A054591, A055246, A062548, A065361, A074940, A081601, A081603, A081611, A083096, A089118, A121153, A170943, A185256.
For generating functions Product_{k>=0} (1+a*x^(b^k)) for the following values of (a,b) see: (1,2) A000012 and A000027, (1,3) A039966 and A005836, (1,4) A151666 and A000695, (1,5) A151667 and A033042, (2,2) A001316, (2,3) A151668, (2,4) A151669, (2,5) A151670, (3,2) A048883, (3,3) A117940, (3,4) A151665, (3,5) A151671, (4,2) A102376, (4,3) A151672, (4,4) A151673, (4,5) A151674.
Row 3 of array A104257.
Summary of increasing sequences avoiding arithmetic progressions of specified lengths (the second of each pair is obtained by adding 1 to the first):
   3-term AP: A005836 (>=0), A003278 (>0);
   4-term AP: A005839 (>=0), A005837 (>0);
   5-term AP: A020654 (>=0), A020655 (>0);
   6-term AP: A020656 (>=0), A005838 (>0);
   7-term AP: A020657 (>=0), A020658 (>0);
   8-term AP: A020659 (>=0), A020660 (>0);
   9-term AP: A020661 (>=0), A020662 (>0);
  10-term AP: A020663 (>=0), A020664 (>0).
See also A000452.

a005836 n = a005836_list !! (n-1)
a005836_list = filter ((== 1) . a039966) [0..]
-- Reinhard Zumkeller, Jun 09 2012, Sep 29 2011

function a(n)
    m, r, b = n, 0, 1
    while m > 0
        m, q = divrem(m, 2)
        r += b * q
        b *= 3
    end
r end; [a(n) for n in 0:57] |> println # Peter Luschny, Jan 03 2021

t := (j, n) -> add(binomial(n,k)^j, k=0..n):
for i from 1 to 400 do
    if(t(4,i) mod 3 <>0) then print(i) fi
od; # Gary Detlefs, Nov 28 2011
# alternative Maple program:
a:= proc(n) option remember: local k, m:
if n=1 then 0 elif n=2 then 1 elif n>2 then k:=floor(log[2](n-1)): m:=n-2^k: procname(m)+3^k: fi: end proc:
seq(a(n), n=1.. 20); # Paul Weisenhorn, Mar 22 2020
# third Maple program:
a:= n-> `if`(n=1, 0, irem(n-1, 2, 'q')+3*a(q+1)):
seq(a(n), n=1..100);  # Alois P. Heinz, Jan 26 2022

Table[FromDigits[IntegerDigits[k, 2], 3], {k, 60}]
Select[Range[0, 400], DigitCount[#, 3, 2] == 0 &] (* Harvey P. Dale, Jan 04 2012 *)
Join[{0}, Accumulate[Table[(3^IntegerExponent[n, 2] + 1)/2, {n, 57}]]] (* IWABUCHI Yu(u)ki, Aug 01 2012 *)
FromDigits[#,3]&/@Tuples[{0,1},7] (* Harvey P. Dale, May 10 2019 *)

A=vector(100);for(n=2,#A,A[n]=if(n%2,3*A[n\2+1],A[n-1]+1));A \\ Charles R Greathouse IV, Jul 24 2012

is(n)=while(n,if(n%3>1,return(0));n\=3);1 \\ Charles R Greathouse IV, Mar 07 2013

a(n) = fromdigits(binary(n-1),3);  \\ Gheorghe Coserea, Jun 15 2018

def A005836(n):
    return int(format(n-1,'b'),3) # Chai Wah Wu, Jan 04 2015

a(n) = A005823(n)/2 = A003278(n)-1 = A033159(n)-2 = A033162(n)-3.
Numbers n such that the coefficient of x^n is > 0 in prod (k >= 0, 1 + x^(3^k)). - Benoit Cloitre, Jul 29 2003
a(n+1) = Sum_{k=0..m} b(k)* 3^k and n = Sum( b(k)* 2^k ).
a(2n+1) = 3a(n+1), a(2n+2) = a(2n+1) + 1, a(0) = 0.
a(n+1) = 3*a(floor(n/2)) + n - 2*floor(n/2). - Ralf Stephan, Apr 27 2003
G.f.: (x/(1-x)) * Sum_{k>=0} 3^k*x^2^k/(1+x^2^k). - Ralf Stephan, Apr 27 2003
a(n) = Sum_{k = 1..n-1} (1 + 3^A007814(k)) / 2. - Philippe Deléham, Jul 09 2005
From Reinhard Zumkeller, Mar 02 2008: (Start)
A081603(a(n)) = 0.
If the offset were changed to zero, then: a(0) = 0, a(n+1) = f(a(n)+1, a(n)+1) where f(x, y) = if x < 3 and x <> 2 then y else if x mod 3 = 2 then f(y+1, y+1) else f(floor(x/3), y). (End)
With offset a(0) = 0: a(n) = Sum_{k>=0} A030308(n,k)*3^k. - Philippe Deléham, Oct 15 2011
a(2^n) = A003462(n). - Philippe Deléham, Jun 06 2015
We have liminf_{n->infinity} a(n)/n^(log(3)/log(2)) = 1/2 and limsup_{n->infinity} a(n)/n^(log(3)/log(2)) = 1. - Gheorghe Coserea, Sep 13 2015
a(2^k+m) = a(m) + 3^k with 1 <= m <= 2^k and 1 <= k, a(1)=0, a(2)=1. - Paul Weisenhorn, Mar 22 2020
Sum_{n>=2} 1/a(n) = 2.682853110966175430853916904584699374821677091415714815171756609672281184705... (calculated using Baillie and Schmelzer's kempnerSums.nb, see Links). - Amiram Eldar, Feb 12 2022
A065361(a(n)) = n-1. - Rémy Sigrist, Feb 06 2023
a(n) ≍ n^k, where k = log 3/log 2 = 1.5849625007. (I believe the constant varies from 1/2 to 1.) - Charles R Greathouse IV, Mar 29 2024

Offset corrected by N. J. A. Sloane, Mar 02 2008

Edited by the Associate Editors of the OEIS, Apr 07 2009

A229037 The "forest fire": sequence of positive integers where each is chosen to be as small as possible subject to the condition that no three terms a(j), a(j+k), a(j+2k) (for any j and k) form an arithmetic progression.

Original entry on oeis.org

1, 1, 2, 1, 1, 2, 2, 4, 4, 1, 1, 2, 1, 1, 2, 2, 4, 4, 2, 4, 4, 5, 5, 8, 5, 5, 9, 1, 1, 2, 1, 1, 2, 2, 4, 4, 1, 1, 2, 1, 1, 2, 2, 4, 4, 2, 4, 4, 5, 5, 8, 5, 5, 9, 9, 4, 4, 5, 5, 10, 5, 5, 10, 2, 10, 13, 11, 10, 8, 11, 13, 10, 12, 10, 10, 12, 10, 11, 14, 20, 13

Offset: 1

Jack W Grahl, Sep 11 2013

Added name "forest fire" to make it easier to locate this sequence. - N. J. A. Sloane, Sep 03 2019
This sequence and A235383 and A235265 were winners in the best new sequence contest held at the OEIS Foundation booth at the 2014 AMS/MAA Joint Mathematics Meetings. - T. D. Noe, Jan 20 2014
See A236246 for indices n such that a(n)=1. - M. F. Hasler, Jan 20 2014
See A241673 for indices n such that a(n)=2^k. - Reinhard Zumkeller, Apr 26 2014
The graph (for up to n = 10000) has an eerie similarity (why?) to the distribution of rising smoke particles subjected to a lateral wind, and where the particles emanate from randomly distributed burning areas in a fire in a forest or field. - Daniel Forgues, Jan 21 2014
The graph (up to n = 100000) appears to have a fractal structure. The dense areas are not random but seem to repeat, approximately doubling in width and height each time. - Daniel Forgues, Jan 21 2014
a(A241752(n)) = n and a(m) != n for m < A241752(n). - Reinhard Zumkeller, Apr 28 2014
The indices n such that a(n) = 1 are given by A236313 (relative spacing) up to 19 terms, and A003278 (directly) up to 20 terms. If just placing ones, the 21st 1 would be n=91. The sequence A003278 fails at n=91 because the numbers filling the gaps create an arithmetic progression with a(27)=9, a(59)=5 and a(91)=1. Additionally, if you look at indices n starting at the first instance of 4 or 5, A003278/A236313 provide possible indices for a(n)=4 or a(n)=5, with some indexes instead filled by numbers < (4,5). A003278/A236313 also fail to predict indices for a(n)=4 or a(n)=5 by the ~20th term. - Daniel Putt, Sep 29 2022

Giovanni Resta, Alois P. Heinz, and Charles R Greathouse IV, Table of n, a(n) for n = 1..100000 (1..1000 from Resta, 1001..10000 from Heinz, and 10001..100000 from Greathouse)
Jean Bickerton, Colored plot of 200000 terms; color is difference from previous term in sequence
Xan Gregg, Enhanced scatterplot of 10000 terms [In this plot, the points have been made translucent to reduce the information lost to overstriking, and the point size varies with n in an attempt to keep the density comparable.]
OEIS, Pin plot of 200 terms and scatterplot of 10000 terms
Sébastien Palcoux, On the first sequence without triple in arithmetic progression (version: 2019-08-21), MathOverflow
Sébastien Palcoux, Table of n, a(n) for n = 1..1000000
Sébastien Palcoux, Density plot of the first 1000000 terms
Reddit User "garnet420", Colored plot of 16 million terms; horizontal divisions are 1000000; vertical divisions are 25000
Reddit User "garnet420", B/W plot of 16 million terms; horizontal divisions are 1000000; vertical divisions are 25000
N. J. A. Sloane, New Gilbreath Conjectures, Sum and Erase, Dissecting Polygons, and Other New Sequences, Doron Zeilberger's Exper. Math. Seminar, Rutgers, Sep 14 2023: Video, Slides, Updates. (Mentions this sequence.)
N. J. A. Sloane and Brady Haran, Amazing Graphs II (including Star Wars), Numberphile video (2019)
Index entries for sequences with interesting graphs or plots
Index entries for non-averaging sequences

Cf. A094870, A362942 (partial sums).
For a variant see A309890.
A selection of sequences related to "no three-term arithmetic progression": A003002, A003003, A003278, A004793, A005047, A005487, A033157, A065825, A092482, A093678, A093679, A093680, A093681, A093682, A094870, A101884, A101886, A101888, A140577, A185256, A208746, A229037.

import Data.IntMap (empty, (!), insert)
a229037 n = a229037_list !! (n-1)
a229037_list = f 0 empty  where
   f i m = y : f (i + 1) (insert (i + 1) y m) where
     y = head [z | z <- [1..],
                   all (\k -> z + m ! (i - k) /= 2 * m ! (i - k `div` 2))
                       [1, 3 .. i - 1]]
-- Reinhard Zumkeller, Apr 26 2014

a[1] = 1; a[n_] := a[n] = Block[{z = 1}, While[Catch[ Do[If[z == 2*a[n-k] - a[n-2*k], Throw@True], {k, Floor[(n-1)/2]}]; False], z++]; z]; a /@ Range[100] (* Giovanni Resta, Jan 01 2014 *)

step(v)=my(bad=List(),n=#v+1,t); for(d=1,#v\2,t=2*v[n-d]-v[n-2*d]; if(t>0, listput(bad,t))); bad=Set(bad); for(i=1,#bad, if(bad[i]!=i, return(i))); #bad+1
first(n)=my(v=List([1])); while(n--, listput(v, step(v))); Vec(v) \\ Charles R Greathouse IV, Jan 21 2014

A229037_list = []
for n in range(10**6):
    i, j, b = 1, 1, set()
    while n-2*i >= 0:
        b.add(2*A229037_list[n-i]-A229037_list[n-2*i])
        i += 1
        while j in b:
            b.remove(j)
            j += 1
    A229037_list.append(j) # Chai Wah Wu, Dec 21 2014

a(n) <= (n+1)/2. - Charles R Greathouse IV, Jan 21 2014

A003002 Size of the largest subset of the numbers [1...n] which does not contain a 3-term arithmetic progression.

Original entry on oeis.org

0, 1, 2, 2, 3, 4, 4, 4, 4, 5, 5, 6, 6, 7, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 10, 10, 11, 11, 11, 11, 12, 12, 13, 13, 13, 13, 14, 14, 14, 14, 15, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 17, 17, 17, 18, 18, 18, 18, 19, 19, 19, 19, 19, 20, 20, 20, 20, 20, 20, 20, 20, 21, 21, 21, 22, 22, 22, 22

Offset: 0

N. J. A. Sloane

"Sequences containing no 3-term arithmetic progressions" is another phrase people may be searching for.
a(n) = size of largest subset of [1..n] such that no term is the average of any two others. These are also called non-averaging sets, or 3-free sequences. - N. J. A. Sloane, Mar 01 2012
More terms of this sequence can be found directly from A065825, because A003002(n) (this sequence) = the integer k such that A065825(k) <= n < A065825(k+1). - Shreevatsa R, Oct 18 2009

			Examples from Dybizbanski (2012) (includes earlier examples found by other people):
n, a(n), example of an optimal subset:
0, 0, []
1, 1, [1]
2, 2, [1, 2]
4, 3, [1, 2, 4]
5, 4, [1, 2, 4, 5]
9, 5, [1, 2, 4, 8, 9]
11, 6, [1, 2, 4, 5, 10, 11]
13, 7, [1, 2, 4, 5, 10, 11, 13]
14, 8, [1, 2, 4, 5, 10, 11, 13, 14]
20, 9, [1, 2, 6, 7, 9, 14, 15, 18, 20]
24, 10, [1, 2, 5, 7, 11, 16, 18, 19, 23, 24]
26, 11, [1, 2, 5, 7, 11, 16, 18, 19, 23, 24, 26]
30, 12, [1, 3, 4, 8, 9, 11, 20, 22, 23, 27, 28, 30]
32, 13, [1, 2, 4, 8, 9, 11, 19, 22, 23, 26, 28, 31, 32]
36, 14, [1, 2, 4, 8, 9, 13, 21, 23, 26, 27, 30, 32, 35, 36]
40, 15, [1, 2, 4, 5, 10, 11, 13, 14, 28, 29, 31, 32, 37, 38, 40]
41, 16, [1, 2, 4, 5, 10, 11, 13, 14, 28, 29, 31, 32, 37, 38, 40, 41]
51, 17, [1, 2, 4, 5, 10, 13, 14, 17, 31, 35, 37, 38, 40, 46, 47, 50, 51]
54, 18, [1, 2, 5, 6, 12, 14, 15, 17, 21, 31, 38, 39, 42, 43, 49, 51, 52, 54]
58, 19, [1, 2, 5, 6, 12, 14, 15, 17, 21, 31, 38, 39, 42, 43, 49, 51, 52, 54, 58]
63, 20, [1, 2, 5, 7, 11, 16, 18, 19, 24, 26, 38, 39, 42, 44, 48, 53, 55, 56, 61, 63]
71, 21, [1, 2, 5, 7, 10, 17, 20, 22, 26, 31, 41, 46, 48, 49, 53, 54, 63, 64, 68, 69, 71]
74, 22, [1, 2, 7, 9, 10, 14, 20, 22, 23, 25, 29, 46, 50, 52, 53, 55, 61, 65, 66, 68, 73, 74]
82, 23, [1, 2, 4, 8, 9, 11, 19, 22, 23, 26, 28, 31, 49, 57, 59, 62, 63, 66, 68, 71, 78, 81, 82]

H. L. Abbott, On a conjecture of Erdos and Straus on non-averaging sets of integers, Proc. 5th British Combinatorial Conference, 1975, pp. 1-4.
Bloom, T. F. (2014). Quantitative results in arithmetic combinatorics (Doctoral dissertation, University of Bristol).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
E. G. Straus, Nonaveraging sets. In Combinatorics (Proc. Sympos. Pure Math., Vol. XIX, Univ. California, Los Angeles, Calif., 1968), pp. 215-222. Amer. Math. Soc., Providence, R.I., 1971. MR0316255 (47 #4803)
T. Tao and V. Vu, Additive Combinatorics, Problem 10.1.3.

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..211
Harvey Abbott, Extremal problems on nonaveraging and nondividing sets, Pacific Journal of Mathematics 91.1 (1980): 1-12.
Harvey Abbott, On the Erdős-Straus non-averaging set problem, Acta Mathematica Hungarica 47.1-2 (1986): 117-119.
Tanbir Ahmed, Janusz Dybizbanski and Hunter Snevily, Unique Sequences Containing No k-Term Arithmetic Progressions, Electronic Journal of Combinatorics, 20(4), 2013, #P29.
F. Behrend, On sets of integers which contain no three terms in an arithmetic progression, Proc. Nat. Acad. Sci. USA, v. 32, 1946, pp. 331-332. MR0018694.
Thomas F. Bloom, A quantitative improvement for Roth's theorem on arithmetic progressions, Journal of the London Mathematical Society, 93(3), 643-663 (2016).
Thomas Bloom, Problem 3, Problem 139, Problem 140, and Problem 142, Erdős Problems.
Thomas F. Bloom and Olof Sisask, Logarithmic bounds for Roth's theorem via almost-periodicity, arXiv:1810.12791 [math.CO], 2018-2019.
Thomas F. Bloom and Olof Sisask, Breaking the logarithmic barrier in Roth's theorem on arithmetic progressions, 2020 preprint. arXiv:2007.03528 [math.NT]
Thomas F. Bloom and Olof Sisask, The Kelley--Meka bounds for sets free of three-term arithmetic progressions, arXiv:2302.07211 [math.NT], 2023.
Fausto A. C. Cariboni, Sets of maximal span that yield a(n) for n = 4..209, Sep 02 2018.
Irene Choi, Shreyas Ekanathan, Aidan Gao, Tanya Khovanova, Sylvia Zia Lee, Rajarshi Mandal, Vaibhav Rastogi, Daniel Sheffield, Michael Yang, Angela Zhao, and Corey Zhao, The Struggles of Chessland, arXiv:2212.01468 [math.HO], 2022.
Janusz Dybizbanski, Sequences containing no 3-term arithmetic progressions, Electronic Journal of Combinatorics, 19(2), 2012, #P15. [Gives first 120 terms].
P. Erdős and E. G. Straus, Nonaveraging sets II, In Combinatorial theory and its applications, II (Proc. Colloq., Balatonfüred, 1969), pp. 405-411. North-Holland, Amsterdam, 1970. MR0316256 (47 #4804).
Zander Kelley and Raghu Meka, Strong Bounds for 3-Progressions, arXiv:2302.05537 [math.NT], 2023-2024.
Zander Kelley, Strong Bounds for 3-Progressions: In-Depth, Institute for Advanced Study, YouTube video, Mar 21, 2023.
Raghu Meka, Strong Bounds for 3-Progressions, Institute for Advanced Study, YouTube video, Mar 20, 2023.
Kevin O'Bryant, Sets of Natural Numbers with Proscribed Subsets, J. Int. Seq. 18 (2015) # 15.7.7.
Quanta Magazine, 2023's Biggest Breakthroughs in Math, Three Arithmetic Progressions, YouTube video, Dec 23, 2023.
Karl C. Rubin, On sequences of integers with no k terms in arithmetic progression, 1973 [Scanned copy, with correspondence]
Tom Sanders, On Roth's theorem on progressions, arXiv:1011.0104 [math.CA], 2010-2011; Annals of Mathematics 174:1 (2011), pp. 619-636.
Z. Shao, F. Deng, M. Liang, and X. Xu, On sets without k-term arithmetic progression, Journal of Computer and System Sciences 78 (2012) 610-618.
N. J. A. Sloane, New Gilbreath Conjectures, Sum and Erase, Dissecting Polygons, and Other New Sequences, Doron Zeilberger's Exper. Math. Seminar, Rutgers, Sep 14 2023: Video, Slides, Updates. (Mentions this sequence.)
Terence Tao, Erdős problem database, see nos. 3, 139-140, 142.
Samuel S. Wagstaff, Jr., On k-free sequences of integers, Math. Comp., 26 (1972), 767-771.
Wikipedia, Salem-Spencer set
Index entries related to non-averaging sequences

The classical lower bound is A104406; A229037 gives a "greedy" lower bound. - N. J. A. Sloane, Apr 29 2023
Cf. A003003, A003004, A003005, A065825, A208746, A143824.
Cf. A358062 (diagonal domination number for the n X n queen graph).
A selection of sequences related to "no three-term arithmetic progression": A003002, A003003, A003278, A004793, A005047, A005487, A033157, A065825, A092482, A093678, A093679, A093680, A093681, A093682, A094870, A101884, A101886, A101888, A140577, A185256, A208746, A229037.

(* Program not suitable to compute a large number of terms *)
a[n_] := a[n] = For[r = Range[n]; k = n, k >= 1, k--, If[AnyTrue[Subsets[r, {k}], FreeQ[#, {_, a_, _, b_, _, c_, _} /; b - a == c - b] &], Return[k]]];
Table[Print["a(", n, ") = ", a[n]]; a[n], {n, 1, 25}] (* Jean-François Alcover, Jan 21 2018 *)

ap3(v)=for(i=1,#v-2, for(j=i+2,#v, my(t=v[i]+v[j]); if(t%2==0 && setsearch(v,t/2), return(1)))); 0
a(N)=forstep(n=N,2,-1, forvec(v=vector(n,i,[1,N]),if(!ap3(v), return(n)),2)); N \\ Charles R Greathouse IV, Apr 22 2022

Sanders proves that a(n) << n*(log log n)^5/log n. - Charles R Greathouse IV, Jan 22 2016

Bloom & Sisask prove that a(n) << n/(log n)^c for some c > 1. - Charles R Greathouse IV, Oct 11 2022

Extended from 53 terms to 80 terms, using a simple brute-force program with some pruning, by Shreevatsa R, Oct 18 2009
See Dybizbanski (2012) for first 120 terms. - N. J. A. Sloane, Dec 17 2013
Edited by N. J. A. Sloane, Apr 12 2016
a(0)=0 prepended by Alois P. Heinz, May 14 2020

A185256 Stanley Sequence S(0,3).

Original entry on oeis.org

0, 3, 4, 7, 9, 12, 13, 16, 27, 30, 31, 34, 36, 39, 40, 43, 81, 84, 85, 88, 90, 93, 94, 97, 108, 111, 112, 115, 117, 120, 121, 124, 243, 246, 247, 250, 252, 255, 256, 259, 270, 273, 274, 277, 279, 282, 283, 286, 324, 327, 328, 331, 333, 336, 337, 340, 351, 354, 355, 358, 360, 363

Offset: 1

N. J. A. Sloane, Mar 19 2011

nonn

Given a finite increasing sequence V = [v_1, ..., v_k] containing no 3-term arithmetic progression, the Stanley Sequence S(V) is obtained by repeatedly appending the smallest term that is greater than the previous term and such that the new sequence also contains no 3-term arithmetic progression.

			After [0, 3, 4, 7, 9] the next term cannot be 10 or we would have the 3-term A.P. 4,7,10; it cannot be 11 because of 7,9,11; but 12 is OK.

R. K. Guy, Unsolved Problems in Number Theory, E10.

Alois P. Heinz, Table of n, a(n) for n = 1..2048
P. Erdos et al., Greedy algorithm, arithmetic progressions, subset sums and divisibility, Discrete Math., 200 (1999), 119-135.
J. L. Gerver and L. T. Ramsey, Sets of integers with no long arithmetic progressions generated by the greedy algorithm, Math. Comp., 33 (1979), 1353-1359.
R. A. Moy, On the growth of the counting function of Stanley sequences, arXiv:1101.0022 [math.NT], 2010-2012.
R. A. Moy, On the growth of the counting function of Stanley sequences, Discrete Math., 311 (2011), 560-562.
A. M. Odlyzko and R. P. Stanley, Some curious sequences constructed with the greedy algorithm, 1978.
S. Savchev and F. Chen, A note on maximal progression-free sets, Discrete Math., 306 (2006), 2131-2133.
Index entries for non-averaging sequences

For other examples of Stanley Sequences see A005487, A005836, A187843, A188052, A188053, A188054, A188055, A188056, A188057.

A092482 Sequence contains no 3-term arithmetic progression, other than its initial terms 1, 2, 3.

Original entry on oeis.org

1, 2, 3, 6, 7, 14, 15, 17, 18, 36, 37, 39, 40, 45, 46, 48, 49, 98, 99, 101, 102, 107, 108, 110, 111, 125, 126, 128, 129, 134, 135, 137, 138, 276, 277, 279, 280, 285, 286, 288, 289, 303, 304, 306, 307, 312, 313, 315, 316, 357, 358, 360, 361, 366, 367, 369, 370

Offset: 1

Ralf Stephan, Apr 04 2004

nonn

a(1)=1, a(2)=2, a(3)=3; a(n) is least k such that no three terms of a(1), a(2), ..., a(n-1), k form an arithmetic progression, except for the first triple (1,2,3).

David A. Corneth, Table of n, a(n) for n = 1..8193 (first 512 terms by Jean-François Alcover)
Eric Weisstein's World of Mathematics, Nonarithmetic Progression Sequence.
Index entries related to non-averaging sequences

Cf. A004793, A033157.

a[n_] := a[n] = If[n < 4, n, For[k = a[n - 1] + 1, True, k++, sp = SequencePosition[Append[Array[a, n - 1], k], {x_, _, y_, _, z_} /; y - x == z - y, 2]; If[sp == {{1, 3}}, Return[k]]]];
Table[Print[n, " ", a[n]]; a[n], {n, 1, 512}]
(* Comparing with data from conjectured formula: *)
b[n_] := If[n < 4, n, 1 + 2^(Length[id = IntegerDigits[n - 2, 2]] - 1) + FromDigits[id, 3]];
Table[b[n], {n, 1, 512}] (* Jean-François Alcover, Jan 15 2019 *)
(* Second [much faster] program: *)
upto[m_] := Module[{n, v, i, j}, n = Max[m, 3]; v = Table[1, {n}]; For[i = 3, i <= n-1, i++, If[v[[i]] == 1, For[j = Max[1, 2i-n], j <= Min[2n-i, i-1], j++, If[v[[j]] == 1, v[[2i-j]] = 0]]]]; Position[v, 1] // Flatten]; upto[12000] (* Jean-François Alcover, Jan 16 2019, after David A. Corneth *)

upto(n) = n=max(n,3); v=vector(n, i, 1); for(i=3, n-1, if(v[i]==1, for(j = max(1, 2*i-n), min(2*n-i,i-1), c=2*i - j; if(v[j]==1, v[2*i-j]=0; )))); select(x -> x==1, v, 1) \\ David A. Corneth, Jan 15 2019

For n > 2, a(n+2) = 1 + 2^floor(log_2(n)) + Sum_{k=1..n} (3^A007814(n) + 1)/2 = 1 + A053644(n) + A005836(n) (conjectured and checked up to n=512).

Name clarified by Charles R Greathouse IV, Jan 30 2014

A093682 Array T(m,n) by antidiagonals: nonarithmetic-3-progression sequences with simple closed forms.

Original entry on oeis.org

1, 2, 1, 4, 3, 1, 5, 4, 4, 1, 10, 6, 5, 7, 1, 11, 10, 8, 8, 10, 1, 13, 12, 10, 10, 11, 19, 1, 14, 13, 13, 11, 13, 20, 28, 1, 28, 15, 14, 16, 14, 22, 29, 55, 1, 29, 28, 17, 17, 20, 23, 31, 56, 82, 1, 31, 30, 28, 20, 22, 28, 32, 58, 83, 163, 1, 32, 31, 31, 28, 23, 29, 37, 59, 85

Offset: 0

Ralf Stephan, Apr 09 2004

The nonarithmetic-3-progression sequences starting with a(1)=1, a(2)=1+3^m or 1+2*3^m, m >= 0, seem to have especially simple 'closed' forms. None of these formulas have been proved, however.

T(m,1)=1, T(m,2) = 1 + (1 + [m even])*3^floor(m/2) = 1 + A038754(m), m >= 0, n > 0; T(m,n) is least k such that no three terms of T(m,1), T(m,2), ..., T(m,n-1), k form an arithmetic progression.

			Array begins:
  1,  2,  4,  5, 10, 11, 13, ...
  1,  3,  4,  6, 10, 12, 13, ...
  1,  4,  5,  8, 10, 13, 14, ...
  1,  7,  8, 10, 11, 16, 17, ...
  1, 10, 11, 13, 14, 20, 22, ...
  ...

Eric Weisstein's World of Mathematics, Nonarithmetic Progression Sequence.
Index entries related to non-averaging sequences

Rows 0-6 are A003278, A004793, A033157, A093678, A093679, A093680, A093681.

Column 2 is 1+A038754. Cf. A092482, A033158.

T(m, n) = (Sum_{k=1..n-1} (3^A007814(k) + 1)/2) + f(n), with f(n) a P-periodic function, where P <= 2^floor((m+3)/2) (conjectured and checked up to m=13, n=1000).

The formula implies that T(m, n) = b(n-1) where b(2n) = 3b(n) + p(n), b(2n+1) = 3b(n) + q(n), with p, q sequences generated by rational o.g.f.s.

A033157 Begins with (1, 4); avoids 3-term arithmetic progressions.

Original entry on oeis.org

1, 4, 5, 8, 10, 13, 14, 17, 28, 31, 32, 35, 37, 40, 41, 44, 82, 85, 86, 89, 91, 94, 95, 98, 109, 112, 113, 116, 118, 121, 122, 125, 244, 247, 248, 251, 253, 256, 257, 260, 271, 274, 275, 278, 280, 283, 284, 287, 325, 328, 329, 332, 334, 337, 338, 341, 352, 355, 356, 359, 361

Offset: 1

N. J. A. Sloane

Also called Stanley Sequence S[1,4]. - Ralf Stephan, Jan 31 2014

F. Iacobescu, 'Smarandache Partition Type and Other Sequences.' Bull. Pure Appl. Sci. 16E, 237-240, 1997.
H. Ibstedt, A Few Smarandache Sequences, Smarandache Notions Journal, Vol. 8, No. 1-2-3, 1997, 170-183.

Eric Weisstein's World of Mathematics, Nonarithmetic Progression Sequence
Index entries related to non-averaging sequences

Cf. A092482, A004793. Row 2 of array in A093682.

Da(n)=if(n<1,1,if(n%2==0,3*Da(n/2)+5-13*((n/2)%2)-6*((n/2)%4==2),3)) /* Ralf Stephan */

Partial sums of Da(n), where Da(n) is defined in the PARI program.
a(n) = A004793(n) + [n is even] + [ceiling(n/2) is even]. Proof by Lawrence Sze. - Ralf Stephan, Nov 15 2004
a(n) = A033161(n) - 1 = A185256(n) + 1. - Ralf Stephan, Jan 31 2014

A003003 Size of the largest subset of the numbers [1...n] which doesn't contain a 4-term arithmetic progression.

Original entry on oeis.org

1, 2, 3, 3, 4, 5, 5, 6, 7, 8, 8, 8, 9, 9, 10, 10, 11, 11, 12, 12, 13, 13, 14, 14, 15, 15, 16, 17, 17, 18, 18, 18, 19, 20, 20, 20, 21, 21, 21, 22, 22, 22, 23, 23, 24, 24, 24, 25, 25, 26, 26, 26, 27, 28, 28, 28, 28, 29, 29, 30, 30, 30, 30, 31, 31, 32, 32, 33, 33, 34, 34, 34

Offset: 1

N. J. A. Sloane

These subsets have been called 4-free sequences.
Szemeredi's theorem for arithmetic progressions of length 4 asserts that a(n) is o(n) as n -> infinity. - Doron Zeilberger, Mar 26 2008
False g.f. (z^12 + 1 - z^11 - z^10 + z^8 - z^6 + z^5 - z^3 + z)/((z+1)*(z-1)^2) was conjectured by Simon Plouffe in his 1992 dissertation, but in fact is wrong (cf. A136746).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Fausto A. C. Cariboni, Table of n, a(n) for n = 1..142
Thomas Bloom, Problem 3, Problem 139, and Problem 142, Erdős Problems.
Fausto A. C. Cariboni, Sets that yield a(n) for n = 5..142, Jun 15 2018
Kevin O'Bryant, Sets of Natural Numbers with Proscribed Subsets, J. Int. Seq. 18 (2015) # 15.7.7
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Karl C. Rubin, On sequences of integers with no k terms in arithmetic progression, 1973 [Scanned copy, with correspondence]
Zehui Shao, Fei Deng, Meilian Liang, and Xiaodong Xu, On sets without k-term arithmetic progression, Journal of Computer and System Sciences 78 (2012) 610-618.
Terence Tao, Erdős problem database, see nos. 3, 139, 142.
Samuel S. Wagstaff, Jr., On k-free sequences of integers, Math. Comp., 26 (1972), 767-771.

Cf. A003002, A003004, A003005, A065825, A136746.

A selection of sequences related to "no three-term arithmetic progression": A003002, A003003, A003278, A004793, A005047, A005487, A033157, A065825, A092482, A093678, A093679, A093680, A093681, A093682, A094870, A101884, A101886, A101888, A140577, A185256, A208746, A229037.

a(52)-a(72) from Rob Pratt, Jul 09 2015

A093678 Sequence contains no 3-term arithmetic progression, starting with 1, 7.

Original entry on oeis.org

1, 7, 8, 10, 11, 16, 17, 20, 28, 34, 35, 37, 38, 43, 44, 47, 82, 88, 89, 91, 92, 97, 98, 101, 109, 115, 116, 118, 119, 124, 125, 128, 244, 250, 251, 253, 254, 259, 260, 263, 271, 277, 278, 280, 281, 286, 287, 290, 325, 331, 332, 334, 335, 340, 341, 344, 352

Offset: 1

Ralf Stephan, Apr 09 2004

a(1)=1, a(2)=7; a(n) is least k such that no three terms of a(1), a(2), ..., a(n-1), k form an arithmetic progression.

Robert Israel, Table of n, a(n) for n = 1..10000
Index entries related to non-averaging sequences

Cf. A004793, A033157, A093679, A093680, A093681, A092482.

Row 3 of array in A093682.

N:= 1000: # to get all terms <= N
V:= Vector(N,1):
A[1]:= 1: A[2]:= 7: k:= 8;
for n from 3 while k < N do
  for k from 1 to n-2 do
    p:= 2*A[n-1]-A[k];
    if p <= N then V[p]:= 0 fi
  od:
  for k from A[n-1]+1 to N do
    if V[k] = 1 then A[n]:= k; nmax:= n; break fi;
  od;
od:
seq(A[i],i=1..nmax); # Robert Israel, May 07 2018

a[n_] := Sum[(1/2)(3^IntegerExponent[k, 2]+1), {k, 1, n-1}] + (1/8)( 12(-1)^n - 7Sin[n Pi/2] + 7Sin[3n Pi/2] - Sin[(n+1)Pi/4] + Sin[(5n+1) Pi/4] + Cos[n Pi/2] + Cos[3n Pi/2] + Cos[n Pi/4] + Cos[3n Pi/4] + Cos[5n Pi/4] + Cos[7n Pi/4] + Cos[(3n+1)Pi/4] - Cos[(7n+1)Pi/4] + 38); Array[a, 60] (* Jean-François Alcover, Mar 22 2019 *)

a(n) = (Sum_{k=1..n-1} (3^A007814(k) + 1)/2) + f(n), with f(n) an 8-periodic function with values {1, 6, 5, 6, 2, 6, 5, 7, ...}, as proved by Lawrence Sze.

A093679 Sequence contains no 3-term arithmetic progression, starting with 1, 10.

Original entry on oeis.org

1, 10, 11, 13, 14, 20, 22, 23, 28, 37, 38, 40, 41, 47, 49, 50, 82, 91, 92, 94, 95, 101, 103, 104, 109, 118, 119, 121, 122, 128, 130, 131, 244, 253, 254, 256, 257, 263, 265, 266, 271, 280, 281, 283, 284, 290, 292, 293, 325, 334, 335, 337, 338, 344, 346, 347

Offset: 1

Ralf Stephan, Apr 09 2004

nonn

a(1)=1, a(2)=10; a(n) is least k such that no three terms of a(1), a(2), ..., a(n-1), k form an arithmetic progression.

Index entries related to non-averaging sequences

Cf. A004793, A033157, A093678, A093680, A093681, A092482.

Row 4 of array in A093682.

a(n) = (Sum_{k=1..n-1} (3^A007814(k) + 1)/2) + f(n), with f(n) an 8-periodic function with values {1, 9, 8, 9, 5, 10, 10, 10, ...}, n >= 1, as proved by Lawrence Sze.

cp's OEIS Frontend

A005836 Numbers whose base-3 representation contains no 2.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Julia

Maple

Mathematica

PARI

PARI

PARI

Python

Formula

Extensions

A229037 The "forest fire": sequence of positive integers where each is chosen to be as small as possible subject to the condition that no three terms a(j), a(j+k), a(j+2k) (for any j and k) form an arithmetic progression.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Python

Formula

A003002 Size of the largest subset of the numbers [1...n] which does not contain a 3-term arithmetic progression.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A185256 Stanley Sequence S(0,3).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

A092482 Sequence contains no 3-term arithmetic progression, other than its initial terms 1, 2, 3.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A093682 Array T(m,n) by antidiagonals: nonarithmetic-3-progression sequences with simple closed forms.

Views

Author

Keywords

Comments

Examples

Links