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 A065825 Smallest k such that n numbers may be picked in {1,...,k} with no three terms in arithmetic progression. 16
 1, 2, 4, 5, 9, 11, 13, 14, 20, 24, 26, 30, 32, 36, 40, 41, 51, 54, 58, 63, 71, 74, 82, 84, 92, 95, 100, 104, 111, 114, 121, 122, 137, 145, 150, 157, 163, 165, 169, 174, 194, 204, 209 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS "Sequences containing no 3-term arithmetic progressions" is another phrase people may be searching for. See also A003002. Don Reble notes large gaps between a(4k) and a(4k+1). Ed Pegg Jr conjectures the 2^k term always equals (3^k+1)/2 and calls these "unprogressive" sets. Jaroslaw Wroblewski (jwr(AT)math.uni.wroc.pl), Nov 04 2003, remarks that this conjecture is known to be false. Further comments from Jaroslaw Wroblewski (jwr(AT)math.uni.wroc.pl), Nov 05 2003: log a(n) / log n tends to 1 was established in 1946 by Behrend. This was extended by me in the Math. Comp. paper. Using appropriately chosen intervals from B(4,9,4) and B(6,9,11) I have determined that log (2a(n)-1) / log n < log 3 / log 2 holds for n=60974 and for n=2^19 since a(60974) <= 19197041, a(524288) <= 515749566. See my web page for further bounds. REFERENCES Knuth, Donald E., Satisfiability,  Fascicle 6, volume 4 of The Art of Computer Programming. Addison-Wesley, 2015, pages 135 and 190, Problem 31. C. R. J. Singleton, "No Progress": Solution to Problem 2472, Journal of Recreational Mathematics, 30(4) 305 1999-2000. LINKS Tanbir Ahmed, Janusz Dybizbanski and Hunter Snevily, Unique Sequences Containing No k-Term Arithmetic Progressions, Electronic Journal of Combinatorics, 20(4), 2013, #P29. F. Behrend, On sets of integers which contain no three terms in an arithmetic progression, Proc. Nat. Acad. Sci. USA, v. 32, 1946, pp. 331-332. Janusz Dybizbanski, Sequences containing no 3-term arithmetic progressions, The Electronic Journal of Combinatorics, 19, no. 2 (2012), #P15. Tom Sanders, On Rothâ€™s theorem on progressions, Annals of Mathematics 174:1 (2011), pp. 619-636; arXiv version, arXiv:1011.0104 [math.CA], 2010-2011. Samuel S. Wagstaff, Jr., On k-free sequences of integers, Math. Comp., 26 (1972), 767-771. J. Wroblewski, A Nonaveraging Set of Integers with a Large Sum of Reciprocals, Math. Comput. 43, 261-262, 1984. J. Wroblewski, Nonaveraging Sets EXAMPLE a(9) = 20 = 1 2 6 7 9 14 15 18 20 a(10) = 24 = 1 2 5 7 11 16 18 19 23 24 a(11) = 26 = 1 2 5 7 11 16 18 19 23 24 26 a(12) = 30 = 1 3 4 8 9 11 20 22 23 27 28 30 (unique) a(13) = 32 = 1 2 4 8 9 11 19 22 23 26 28 31 32 a(14) = 36 = 1 2 4 8 9 13 21 23 26 27 30 32 35 36 a(15) = 40 = 1 2 4 5 10 11 13 14 28 29 31 32 37 38 40 a(16) = 41 = 1 2 4 5 10 11 13 14 28 29 31 32 37 38 40 41 a(17) = 51 = 1 2 4 5 10 13 14 17 31 35 37 38 40 46 47 50 51 a(18) = 54 = 1 2 5 6 12 14 15 17 21 31 38 39 42 43 49 51 52 54 a(19) = 58 = 1 2 5 6 12 14 15 17 21 31 38 39 42 43 49 51 52 54 58 a(20) = 63 = 1 2 5 7 11 16 18 19 24 26 38 39 42 44 48 53 55 56 61 63 a(21) = 71 = 1 2 5 7 10 17 20 22 26 31 41 46 48 49 53 54 63 64 68 69 71 a(22) = 74 = 1 2 7 9 10 14 20 22 23 25 29 46 50 52 53 55 61 65 66 68 73 74 a(23) = 82 = 1 2 4 8 9 11 19 22 23 26 28 31 49 57 59 62 63 66 68 71 78 81 82 a(24) = 84 = 1 3 4 8 9 16 18 21 22 25 30 37 48 55 60 63 64 67 69 76 77 81 82 84 a(25) = 92 = 1 2 6 8 9 13 19 21 22 27 28 39 58 62 64 67 68 71 73 81 83 86 87 90 92 a(26) = 95 = 1 2 4 5 10 11 22 23 25 26 31 32 55 56 64 65 67 68 76 77 82 83 91 92 94 95 a(27) = 100 = 1 3 6 7 10 12 20 22 25 26 29 31 35 62 66 68 71 72 75 77 85 87 90 91 94 96 100 MATHEMATICA ThreeAPFree[n_, k_, a_] := Module[{d, j},    For[d = 1, d < k/2, d ++,     For[j = 1, j <= n - 2, j++,      If[MemberQ[a, a[[j]] + d] && MemberQ[a, a[[j]] + 2 d],       Return[False]]]];    Return[True]]; A065825[n_] := Module[{k, x, a},   k = n;   While[True,    x = Subsets[Range[k], {n}];    For[i = 1, i <= Length[x], i++,     a = x[[i]];     If[a[[1]] != 1 || a[[n]] != k, Continue[]];     If[ThreeAPFree[n, k, a], Return[k]]];    k++]] Table[A065825[n], {n, 1, 10}]  (* Robert Price, Mar 11 2019 *) CROSSREFS Cf. A003002 (three-free sequences), A003278, A003003, A003004, A003005, A225745. a(n) = A005047(n) + 1. - Rob Pratt, Jul 09 2015 Sequence in context: A327185 A047378 A101155 * A113755 A124254 A192615 Adjacent sequences:  A065822 A065823 A065824 * A065826 A065827 A065828 KEYWORD hard,nice,more,nonn AUTHOR Ed Pegg Jr, Nov 23 2001 EXTENSIONS a(19) found by Guenter Stertenbrink in response to an A003278-based puzzle on www.mathpuzzle.com More terms from Don Reble, Nov 25 2001 Five more terms from William Rex Marshall, Mar 24 2002 137 from William Rex Marshall, Nov 15 2003 One more term from William Rex Marshall, Jan 24 2004 a(35)-a(36) (found by Gavin Theobold in 2004) communicated by William Rex Marshall, Mar 10 2007 Added a(37)-a(41) (from Wroblewski's web page), Joerg Arndt, Apr 25 2012 a(42)-a(43) from Fausto A. C. Cariboni, Sep 02 2018 STATUS approved

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Last modified September 22 05:47 EDT 2019. Contains 327287 sequences. (Running on oeis4.)