A229037 -id:A229037 - cp's OEIS frontend

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

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

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

A094870 a(1)=1; for n > 1, a(n) is the minimal positive integer t not equal to a(1), ..., a(n-1) such that t - a(n-i) is not equal to a(n-i) - a(n-2i) for all 1 <= i < n/2.

Original entry on oeis.org

1, 2, 4, 3, 5, 6, 8, 7, 10, 9, 13, 12, 14, 11, 17, 16, 22, 15, 23, 18, 21, 20, 25, 24, 26, 19, 28, 27, 29, 36, 32, 31, 33, 39, 38, 34, 41, 30, 37, 35, 44, 48, 42, 40, 43, 50, 46, 52, 47, 45, 54, 49, 56, 58, 57, 51, 61, 53, 59, 63, 60, 68, 64, 62, 70, 55, 65, 67, 73, 69, 83, 76

Offset: 1

Alec Mihailovs (alec(AT)mihailovs.com), Jun 16 2004

3n/8 <= a(n) < 3n/2 (P. Hegarty).
Conjecture: lim_{n->infinity} a(n)/n = 1 (P. Hegarty).
The Hegarty paper shows that this is a permutation. - Franklin T. Adams-Watters, May 26 2014
Graphically, the sequence {a(n)-n} resembles A293862 (see illustration in Links section). - Rémy Sigrist, Feb 06 2020

			a(3)=4 because it can't be 1=a(1), 2=a(2) and 3=2*a(3-1)-a(3-2).

Carl R. White, Table of n, a(n) for n = 1..10000
Peter Hegarty, Permutations avoiding arithmetic patterns, The Electronic Journal of Combinatorics, 11 (2004), #R39.
Rémy Sigrist, Scatterplot of (n, a(n)-n) for n = 1..500000
Index entries related to non-averaging sequences

Cf. A095689 (inverse permutation), A229037, A293862.

A:=proc(n) option remember; local t, S, i; S:={$1..1000} minus {seq(A(i),i=1..n-1)}; t:=min(S[]); i:=1; while i

a[n_] := a[n] = Module[{t, S, i}, S = Union[Range[1000] ~Complement~ Array[a, n-1]]; t = Min[S]; i = 1; While[i < Floor[(n+1)/2], If[t-a[n-i] == a[n-i] - a[n-2i], S = S ~Complement~ {t}; t = Min[S]; i = 1, i++]]; t]; a[1] = 1;
Table[a[n], {n, 1, 200}] (* Jean-François Alcover, Sep 20 2019, from Maple *)

A100480 a(1)=1 and, for n>1, a(n) is the smallest positive integer such that the subset S(c) of {1,2,3,...,n} defined by S(c)={k|1<=k<=n; a(k)=c} does not contain a 3-term arithmetic progression for any integer c.

Original entry on oeis.org

1, 1, 2, 1, 1, 2, 2, 3, 3, 1, 1, 2, 1, 1, 2, 2, 3, 3, 2, 3, 3, 4, 4, 5, 4, 4, 5, 1, 1, 2, 1, 1, 2, 2, 3, 3, 1, 1, 2, 1, 1, 2, 2, 3, 3, 2, 3, 3, 4, 4, 5, 4, 4, 5, 2, 3, 3, 4, 4, 5, 4, 4, 5, 5, 6, 6, 5, 6, 6, 7, 7, 8, 5, 6, 6, 5, 6, 6, 7, 7, 8, 1, 1, 2, 1, 1, 2, 2, 3, 3, 1, 1, 2, 1, 1, 2, 2, 3, 3, 2, 3, 3, 4, 4, 5

Offset: 1

John W. Layman, Nov 22 2004

nonn

The sequence of values of n for which a(n)=1 is given by {A005836(i)-1}.

Samuel B. Reid, Table of n, a(n) for n = 1..10000
Samuel B. Reid, Python program for A100480

Cf. A005836, A006997, A135415, A229037.

Python
```
# See Links section.
```

a(n + 1) = A006997(n) + 1 for any n >= 0. - Rémy Sigrist, Jun 28 2022

A309890 Lexicographically earliest sequence of positive integers without triples in weakly increasing 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, 2, 4, 4, 5, 5, 10, 5, 5, 10, 10, 11, 13, 10, 11, 10, 11, 13, 10, 10, 12, 13, 10, 13, 11, 12, 20, 11, 1, 1, 2, 1, 1, 2, 2, 4, 4, 1, 1, 2, 1, 1, 2, 2, 4, 4, 2

Offset: 1

Sébastien Palcoux, Aug 21 2019

Formal definition: lexicographically earliest sequence of positive integers a(n) such that for any i > 0, there is no n > 0 such that 2a(n+i) = a(n) + a(n+2i) AND a(n) <= a(n+i) <= a(n+2i).
Sequence suggested by Richard Stanley as a variant of A229037. They differ from the 55th term. The numbers n for which a(n) = 1 are given by A003278, or equally by A005836 (Richard Stanley).
The sequence defined by c(n) = 1 if a(n) = 1 and otherwise c(n) = 0 is A039966 (although with a different offset). - N. J. A. Sloane, Dec 01 2019
Pleasant to listen to (button above) with Instrument no. 13: Marimba (and for better listening, save and convert to MP3).

Sébastien Palcoux, Table of n, a(n) for n = 1..100000
Sébastien Palcoux, On the first sequence without triple in arithmetic progression (version: 2019-08-21), second part, MathOverflow
Sébastien Palcoux, Table of n, a(n) for n = 1..1000000
Sébastien Palcoux, Density plot of the first 1000000 terms

Cf. A003278, A005836, A039966, A229037.

from itertools import count, islice
def A309890_gen(): # generator of terms
    blist = []
    for n in count(0):
        i, j, b = 1, 1, set()
        while n-(i<<1) >= 0:
            x, y = blist[n-2*i], blist[n-i]
            z = (y<<1)-x
            if x<=y<=z:
                b.add(z)
                while j in b:
                    j += 1
            i += 1
        blist.append(j)
        yield j
A309890_list = list(islice(A309890_gen(),30)) # Chai Wah Wu, Sep 12 2023

# %attach SAGE/ThreeFree.spyx
from sage.all import *
cpdef ThreeFree(int n):
     cdef int i,j,k,s,Li,Lj
     cdef list L,Lb
     cdef set b
     L=[1,1]
     for k in range(2,n):
          b=set()
          for i in range(k):
               if 2*((i+k)/2)==i+k:
                    j=(i+k)/2
                    Li,Lj=L[i],L[j]
                    s=2*Lj-Li
                    if s>0 and Li<=Lj:
                         b.add(s)
          if 1 not in b:
               L.append(1)
          else:
               Lb=list(b)
               Lb.sort()
               for t in Lb:
                    if t+1 not in b:
                         L.append(t+1)
                         break
     return L

A236335 Lexicographically earliest sequence of positive integers whose graph has no three collinear points.

Original entry on oeis.org

1, 1, 2, 2, 5, 4, 9, 3, 3, 6, 8, 5, 6, 9, 17, 4, 8, 15, 13, 24, 17, 13, 26, 32, 14, 7, 12, 29, 12, 18, 10, 10, 23, 35, 7, 16, 14, 30, 24, 23, 30, 46, 27, 20, 52, 15, 25, 40, 29, 40, 19, 38, 58, 18, 39, 42, 16, 69, 33, 25, 67, 43, 11, 51, 28, 11, 54, 73, 26, 27

Offset: 1

Tanya Khovanova, Jan 22 2014

nonn

An integer can't appear more than twice in the sequence, which means the sequence tends to infinity.

An increasing version of this sequence is A236336.

			Consider a(5). The previous terms are 1,1,2,2. The value of a(5) can't be 1 because points (1,1),(2,1),(5,1) (corresponding to values a(1), a(2), a(5)) are on the same line: y=1. Points (3,2),(4,2),(5,2) are on the same line y=2, so a(5) can't be 2. Points (1,1),(3,2),(5,3) are on the same line: y=x/2+1/2, so a(5) can't be 3. Points (2,1),(3,2),(5,4) are on the same line: y=x-1, so a(5) can't be 4. Thus a(5)=5.

Grant Garcia, Table of n, a(n) for n = 1..10000
Dániel T. Nagy, Zoltán Lóránt Nagy, and Russ Woodroofe, The extensible No-Three-In-Line problem, arXiv:2209.01447 [math.CO], 2022.

Cf. A229037, A185256, A231334, A236266, A236336, A300002.

b[1] = 1;
b[n_] := b[n] =
  Min[Complement[Range[100],
    Select[Flatten[
      Table[b[k] + (n - k) (b[j] - b[k])/(j - k), {k, n - 2}, {j,
        k + 1, n - 1}]], IntegerQ[#] &]]]
Table[b[k], {k, 70}]

a(n) = A236266(n-1) + 1. - Alois P. Heinz, Jan 23 2014

A361933 Lexicographically earliest sequence of positive integers such that no three terms a(j), a(j+k), a(j+2k) (for any j and k) form an arithmetic progression in any order.

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, 9, 4, 2, 5, 11, 2, 2, 4, 1, 1, 5, 1, 1, 10, 2, 2, 4, 1, 1, 4, 4, 10, 10, 4, 8, 10, 10, 2, 4, 1, 2, 5, 4, 10, 10, 4, 2, 8, 8, 5, 8, 5, 13, 13, 17, 5, 13, 2, 11, 17, 10, 10, 13, 13

Offset: 1

Neal Gersh Tolunsky, Mar 30 2023

nonn

First differs from A229037 and A309890 at a(28).
This sequence avoids all six of the six permutations of a set of three integers in arithmetic progression. For example, the set {1,2,3} can be ordered as tuples (1, 2, 3), (1, 3, 2), (2, 1, 3), (2, 3, 1), (3, 1, 2), and (3, 2, 1).
This sequence is part of a family of variants avoiding different permutations of arithmetic progressions at indices in arithmetic progression:
- A100480 (offset 1), A006997 (offset 0): Prohibits 1,1,1 and progressions of common difference 0.
- A309890: Prohibits 1,2,3 or progressions of the form c, c+d, c+2d, for all d >= 0.
- A373111: Prohibits 1,3,2 or progressions of the form c, c+2d, c+d, for all d >= 0.
- A371457: Prohibits 2,1,3 or progressions of the form c, c-d, c+d, for all d >= 0.
- A371632: Prohibits 2,3,1 or progressions of the form c, c+d, c-d, for all d >= 0.
- A373010: Prohibits 3,1,2 or progressions of the form c, c-2d, c-d, for all d>=0.
- A373052: Prohibits 3,2,1 or progressions of the form c, c-d, c-2d, for all d>=0.
With the sequences prohibiting the six permutations above, there are a total of 64 sequences which prohibit some combination of these six permutations of an arithmetic progression. At least two more of these are in the OEIS:
- A229037 ("forest fire sequence"): Prohibits (progressions of the same general form as) 1,2,3 and 3,2,1 .
- A361933 (the present sequence): Prohibits all six permutations.

			a(28) cannot be 1 because then a(26)=5, a(27)=9, and a(28)=1 could be rearranged to form an arithmetic progression (1, 5, 9). The numbers 2-8 could also create an arithmetic progression so a(28)=9.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program
Index entries for non-averaging sequences
Neal Gersh Tolunsky, Graph of the first 200000 terms

Cf. A229037, A309890.

PARI
```
\\ See Links section.
```

a(n) <= (n+1)/2.

A101886 Smallest natural number sequence without any length 4 equidistant arithmetic subsequences.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 9, 10, 11, 14, 15, 16, 18, 19, 20, 22, 24, 27, 28, 29, 31, 32, 35, 36, 37, 39, 41, 42, 43, 47, 48, 50, 51, 53, 55, 58, 60, 61, 63, 65, 66, 68, 70, 71, 72, 77, 78, 80, 82, 85, 86, 87, 89, 90, 91, 94, 95, 96, 98, 99, 100, 102, 103, 104, 107, 109, 110, 111, 114

Offset: 1

Douglas Stones (dssto1(AT)student.monash.edu.au), Dec 20 2004

nonn

			4 is out because of 1,2,3,4. 13 is out because of 1,5,9,13.

Cf. A101887, A101884, A101888.

A101888 Smallest natural number sequence without any length 5 equidistant arithmetic subsequences.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 13, 14, 16, 17, 18, 19, 22, 23, 24, 25, 27, 28, 29, 30, 32, 33, 34, 35, 37, 38, 39, 40, 43, 44, 45, 46, 48, 49, 50, 51, 53, 54, 55, 56, 58, 59, 60, 61, 64, 65, 66, 67, 69, 70, 71, 72, 74, 75, 76, 77, 79, 80, 81, 82, 86, 87, 88, 90, 91, 92, 93, 95

Offset: 1

Douglas Stones (dssto1(AT)student.monash.edu.au), Dec 20 2004

nonn

			5 is out because of 1,2,3,4,5. 21 is out because of 1,6,11,16,21.

Cf. A101889, A101884, A101886.

A276204 a(0) = a(1) = 0. For n>1 a(n) is the smallest nonnegative integer such that there is no arithmetic progression k,m,n (of length 3) such that a(k)+a(m) = a(n).

Original entry on oeis.org

0, 0, 1, 0, 0, 1, 2, 2, 1, 0, 0, 1, 0, 0, 1, 2, 2, 1, 4, 4, 1, 4, 4, 1, 2, 2, 1, 0, 0, 1, 0, 0, 1, 2, 2, 1, 0, 0, 1, 0, 0, 1, 2, 2, 1, 4, 4, 1, 4, 4, 1, 2, 2, 1, 7, 7, 1, 7, 7, 1, 2, 2, 1, 7, 7, 1, 7, 7, 1, 2, 2, 1, 4, 4, 1, 4, 4, 1, 2, 2, 1, 0, 0, 1, 0, 0, 1, 2, 2, 1, 0, 0, 1, 0, 0, 1, 2, 2, 1, 4, 4, 1, 4, 4, 1, 2, 2, 1, 0, 0, 1, 0, 0, 1, 2, 2, 1, 0, 0, 1, 0

Offset: 0

Michal Urbanski, Aug 24 2016

The sequence has the same set of values as A033627.
The sequence has a kind of a "triple rhythm", compare the distribution of zeros to the Cantor set.
Conjecture 1:
One can calculate a(n) in a following, non-recursive way, using the ternary representation of n.
Let n>=0 be an integer. We consider two cases:
1. There is no digit 2 in the ternary representation of n. Then a(n) = 0.
2. There is a digit 2 in the ternary representation of n.
Let i be the number of the position (counting from right) of the rightmost digit 2 in ternary representation of n, then a(n) = A033627(i).
For example: let n = 19. The ternary representation of 19 is 201. The rightmost digit 2 in the number 201 is on the third position (counting from right), so a(19) = A033627(3) = 4.
Conjecture 2:
The sequence can be generated in a following way:
Start with a zero. Take three consecutive copies of all you have, replace all zeros in the third copy with the next value of A033627, repeat.
Both of these conjectures can be generalized for similarly defined sequences where the length of the arithmetic progression in the definition (in A276204 it is 3) is a prime number, see A276206.
The assumption about primality is essential, for complex lengths of the arithmetic progression the sequence is irregular, see A276205.

			For n = 6 we have that:
a(6)>0, because a(0)+a(3)=0 and 0,3,6 is an arithmetic progression.
a(6)>1, because a(4)+a(5)=0 and 4,5,6 is an arithmetic progression.
There is no such arithmetic progression k,m,6 that a(k)+a(m) = 2, so a(6) = 2.

Michal Urbanski, Table of n, a(n) for n = 0..199999

Cf. A276205 (length 4), A276206 (length 5), A276207 (any length).

Cf. A033627, A229037.

cp's OEIS Frontend

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

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

Views

Author

Keywords

Comments

References

Links

Crossrefs

Extensions

A094870 a(1)=1; for n > 1, a(n) is the minimal positive integer t not equal to a(1), ..., a(n-1) such that t - a(n-i) is not equal to a(n-i) - a(n-2i) for all 1 <= i < n/2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

A100480 a(1)=1 and, for n>1, a(n) is the smallest positive integer such that the subset S(c) of {1,2,3,...,n} defined by S(c)={k|1<=k<=n; a(k)=c} does not contain a 3-term arithmetic progression for any integer c.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Python

Formula

A309890 Lexicographically earliest sequence of positive integers without triples in weakly increasing arithmetic progression.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Python

SageMath

A236335 Lexicographically earliest sequence of positive integers whose graph has no three collinear points.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A361933 Lexicographically earliest sequence of positive integers such that no three terms a(j), a(j+k), a(j+2k) (for any j and k) form an arithmetic progression in any order.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A101886 Smallest natural number sequence without any length 4 equidistant arithmetic subsequences.

Views

Author