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A003002 Size of the largest subset of the numbers [1...n] which does not contain a 3-term arithmetic progression.
(Formerly M0275)
21
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

"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). [R. Shreevatsa (shreevatsa.public(AT)gmail.com), Oct 18 2009]

REFERENCES

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.

Abbott, Harvey. "Extremal problems on nonaveraging and nondividing sets." Pacific Journal of Mathematics 91.1 (1980): 1-12.

Abbott, H. "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.

Behrend, F. A., On sets of integers which contain no three terms in arithmetical progression. Proc. Nat. Acad. Sci. U. S. A. 32, (1946). 331-332. MR0018694.

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)

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.

S. S. Wagstaff, Jr., On k-free sequences of integers, Math. Comp., 26 (1972), 767-771.

LINKS

Table of n, a(n) for n=1..77.

Janusz Dybizbanski, Sequences containing no 3-term arithmetic progressions, Electronic Journal of Combinatorics, 19(2), 2012, #P15. [Gives first 120 terms.

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, Annals of Mathematics 174:1 (2011), pp. 619-636.

Z. Shao, F. Deng, M. Liang, X. Xu, On sets without k-term arithmetic progression, Journal of Computer and System Sciences 78 (2012) 610-618.

Index entries related to non-averaging sequences

FORMULA

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

EXAMPLE

Examples from Dybizbanski (2012) (includes earlier examples found by other people):

n, a(n), example of an optimal subset:

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]

CROSSREFS

Cf. A003003, A003004, A003005, A065825, A208746, A143824.

Sequence in context: A081611 A081228 A270826 * A087180 A029121 A161205

Adjacent sequences:  A002999 A003000 A003001 * A003003 A003004 A003005

KEYWORD

nonn,nice

AUTHOR

N. J. A. Sloane

EXTENSIONS

Extended from 53 terms to 80 terms, using a simple brute-force program with some pruning, by R. Shreevatsa (shreevatsa.public(AT)gmail.com), 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

STATUS

approved

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Last modified October 24 06:43 EDT 2017. Contains 293836 sequences.