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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; 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]
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
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 * A356988 A248186 A248183
KEYWORD
nonn
AUTHOR
EXTENSIONS
a(52)-a(72) from Rob Pratt, Jul 09 2015
STATUS
approved

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Last modified March 28 18:04 EDT 2024. Contains 371254 sequences. (Running on oeis4.)