

A003003


Size of the largest subset of the numbers [1...n] which doesn't contain a 4term 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
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OFFSET

1,2


COMMENTS

These subsets have been called 4free 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)*(z1)^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, 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 kterm arithmetic progression, Journal of Computer and System Sciences 78 (2012) 610618.
S. S. Wagstaff, Jr., On kfree sequences of integers, Math. Comp., 26 (1972), 767771.


CROSSREFS

Cf. A003002, A003004, A003005, A065825, A136746.
A selection of sequences related to "no threeterm 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



