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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

Fausto A. C. Cariboni, Table of n, a(n) for n = 1..142

Fausto A. C. Cariboni, Sets that yield a(n) for n = 5..142., Jun 15 2018

K. 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.

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, 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.

Sequence in context: A067022 A113818 A136746 * A248186 A248183 A049474

Adjacent sequences:  A003000 A003001 A003002 * A003004 A003005 A003006

KEYWORD

nonn

AUTHOR

N. J. A. Sloane

EXTENSIONS

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

STATUS

approved

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Last modified August 14 17:44 EDT 2018. Contains 313751 sequences. (Running on oeis4.)