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A003005 Size of the largest subset of the numbers [1..n] which doesn't contain a 6-term arithmetic progression.

%I A003005 M0459 #54 Sep 03 2025 15:13:51
%S A003005 1,2,3,4,5,5,6,7,8,9,9,10,11,12,13,13,14,15,16,17,17,18,19,20,21,22,
%T A003005 22,22,23,23,23,24,25,25,26,27,28,28,29,30,31,31,31,32,33,34,34,35,36,
%U A003005 37,38,38,38,39,39,40,40,41,42,42,43,44,44,45,46,47,47,48,48
%N A003005 Size of the largest subset of the numbers [1..n] which doesn't contain a 6-term arithmetic progression.
%C A003005 These subsets have been called 6-free sequences.
%D A003005 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A003005 Fausto A. C. Cariboni, <a href="/A003005/b003005.txt">Table of n, a(n) for n = 1..147</a>
%H A003005 Thomas Bloom, <a href="https://www.erdosproblems.com/3">Problem 3</a>, <a href="https://www.erdosproblems.com/139">Problem 139</a>, and <a href="https://www.erdosproblems.com/142">Problem 142</a>, Erdős Problems.
%H A003005 Fausto A. C. Cariboni, <a href="/A003005/a003005.txt">Sets that yield a(n) for n = 7..147</a>, May 20 2018.
%H A003005 Kevin O'Bryant, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/OBryant/obr3.html">Sets of Natural Numbers with Proscribed Subsets</a>, J. Int. Seq. 18 (2015) # 15.7.7.
%H A003005 Karl C. Rubin, <a href="/A003002/a003002.pdf">On sequences of integers with no k terms in arithmetic progression</a>, 1973. [Scanned copy, with correspondence]
%H A003005 Zehui Shao, Fei Deng, Meilian Liang, and Xiaodong Xu, <a href="http://dx.doi.org/10.1016/j.jcss.2011.09.003">On sets without k-term arithmetic progression</a>, Journal of Computer and System Sciences 78 (2012) 610-618.
%H A003005 Terence Tao, <a href="https://github.com/teorth/erdosproblems/blob/main/README.md#table">Erdős problem database</a>, see nos. 3, 139, 142.
%H A003005 Samuel S. Wagstaff, Jr., <a href="http://dx.doi.org/10.1090/S0025-5718-1972-0325500-5">On k-free sequences of integers</a>, Math. Comp., 26 (1972), 767-771.
%Y A003005 Cf. A003002, A003003, A003004, A065825.
%K A003005 nonn,changed
%O A003005 1,2
%A A003005 _N. J. A. Sloane_