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A003004 Size of the largest subset of the numbers [1..n] which does not contain a 5-term arithmetic progression.

%I A003004 M0451 #50 Sep 03 2025 11:40:52
%S A003004 1,2,3,4,4,5,6,7,8,8,9,10,11,12,12,13,14,15,16,16,16,16,16,17,18,18,
%T A003004 19,20,21,21,22,22,23,24,24,25,26,27,28,28,29,30,31,32,32,32,32,32,33,
%U A003004 33,34,35,35,36,36,37,38,39,40,40,41,42,43,44,44,45,46,47,48,48
%N A003004 Size of the largest subset of the numbers [1..n] which does not contain a 5-term arithmetic progression.
%C A003004 These subsets have been called 5-free sequences.
%D A003004 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A003004 Fausto A. C. Cariboni, <a href="/A003004/b003004.txt">Table of n, a(n) for n = 1..115</a>
%H A003004 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 A003004 Fausto A. C. Cariboni, <a href="/A003004/a003004.txt">Sets that yield a(n) for n = 6..115</a>, Apr 30 2018.
%H A003004 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 A003004 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 A003004 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 A003004 Terence Tao, <a href="https://github.com/teorth/erdosproblems/blob/main/README.md#table">Erdős problem database</a>, see nos. 3, 139, 142.
%H A003004 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 A003004 Cf. A003002, A003003, A003005, A065825.
%K A003004 nonn,changed
%O A003004 1,2
%A A003004 _N. J. A. Sloane_
%E A003004 a(51) and beyond from _Fausto A. C. Cariboni_, Apr 30 2018