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A065825
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Smallest k such that n numbers may be picked in {1,...,k} with no three terms in arithmetic progression.
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6
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| 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 yesterday 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.
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REFERENCES
| 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.
C. R. J. Singleton, "No Progress": Solution to Problem 2472, Journal of Recreational Mathematics, 30(4) 305 1999-2000.
S. 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, Mathematics of Computation, Volume 43, Number 167, 1984, pp. 261-262.
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LINKS
| J. Wroblewski, Nonaveraging Sets
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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
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CROSSREFS
| Cf. A003002 (three-free sequences), A003278, A003003, A003004, A003005.
Sequence in context: A069001 A047378 A101155 * A124254 A192615 A065514
Adjacent sequences: A065822 A065823 A065824 * A065826 A065827 A065828
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KEYWORD
| hard,nice,nonn
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AUTHOR
| Ed Pegg Jr (ed(AT)mathpuzzle.com), Nov 23 2001
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EXTENSIONS
| a(19) found by Guenter Stertenbrink in response to an A003278-based puzzle on www.mathpuzzle.com
More terms from Don Reble (djr(AT)nk.ca), Nov 25 2001
Five more terms from William Rex Marshall (w.r.marshall(AT)actrix.co.nz), Mar 24 2002
137 from William Rex Marshall (w.r.marshall(AT)actrix.co.nz), Nov 15 2003
One more term from William Rex Marshall (w.r.marshall(AT)actrix.co.nz), Jan 24 2004
a(35) and a(36) (found by Gavin Theobold in 2004) communicated by William Rex Marshall (w.r.marshall(AT)actrix.co.nz), Mar 10 2007.
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