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A003003 Size of the largest subset of the numbers [1...n] which doesn't contain a 4-term arithmetic progression.
(Formerly M0439)
12
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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).

REFERENCES

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

LINKS

Table of n, a(n) for n=1..72.

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.

Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992.

Karl C. Rubin, On sequences of integers with no k terms in arithmetic progression, 1973 [Scanned copy, with correspondence]

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.

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

CROSSREFS

Cf. A003002, A003004, A003005, A065825.

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.

Sequence in context: A067022 A113818 A136746 * A248186 A248183 A049474

Adjacent sequences:  A003000 A003001 A003002 * A003004 A003005 A003006

KEYWORD

nonn,more

AUTHOR

N. J. A. Sloane.

EXTENSIONS

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

STATUS

approved

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Last modified February 19 22:04 EST 2018. Contains 299357 sequences. (Running on oeis4.)