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 A030126 Schur's numbers (version 1). 2
 2, 5, 14, 45, 161 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS Smallest number such that for any n-coloring of the integers 1, ..., a(n) no color is sum-free, that is, some color contains a triple x + y = z. - Charles R Greathouse IV, Jun 11 2013 REFERENCES R. K. Guy, Unsolved Problems in Number Theory, E11 and E12. Marijn J. H. Heule, Schur Number Five, AAAI 2018. LINKS Leonard D. Baumert and Solomon W. Golomb, Backtrack Programming, Journal of the ACM 12:4 (1965), pp. 516-524. Marijn J. H. Heule, Schur Number Five, arXiv:1711.08076 [cs.LO], 2017. Eric Weisstein's World of Mathematics, Schur Number EXAMPLE Baumert & Golomb find a(4) = 45 and give this example: A = {1, 3, 5, 15, 17, 19, 26, 28, 40, 42, 44} B = {2, 7, 8, 18, 21, 24, 27, 37, 38, 43} C = {4, 6, 13, 20, 22, 23, 25, 30, 32, 39, 41} D = {9, 10, 11, 12, 14, 16, 29, 31, 33, 34, 35, 36} which demonstrates that a(4) > 44. Note that the union of these sets is {1, ..., 44} and none of the sets contains three numbers (perhaps not all distinct) such that one is the sum of the other two. - Charles R Greathouse IV, Jun 11 2013 From Marijn Heule, Dec 12 2017: (Start) Exoo computed the first certificate showing that a(5) > 160: A = {1, 6, 10, 18, 21, 23, 26, 30, 34, 38, 43, 45, 50, 54, 65, 74, 87, 96, 107, 111, 116, 118, 123, 127, 131, 135, 138, 140, 143, 151, 155, 160} B = {2, 3, 8, 14, 19, 20, 24, 25, 36, 47, 51, 62, 73, 88, 99, 110, 114, 125, 136, 137, 141, 142, 147, 153, 158, 159} C = {4, 5, 15, 16, 22, 28, 29, 40, 41, 42, 48, 49, 59, 102, 112, 113, 119, 120, 121, 132, 133, 139, 145, 146, 156, 157} D = {7, 9, 11, 12, 13, 17, 27, 31, 32, 33, 35, 37, 53, 56, 57, 61, 79, 82, 100, 104, 105, 108, 124, 126, 128, 129, 130, 134, 144, 148, 149, 150, 152, 154} E = {44, 52, 55, 58, 60, 63, 64, 66, 67, 68, 69, 70, 71, 72, 75, 76, 77, 78, 80, 81, 83, 84, 85, 86, 89, 90, 91, 92, 93, 94, 95, 97, 98, 101, 103, 106, 109, 117} (End) CROSSREFS Cf. A045652. Sequence in context: A149892 A149893 A149894 * A081444 A268004 A119429 Adjacent sequences:  A030123 A030124 A030125 * A030127 A030128 A030129 KEYWORD nonn,hard,nice,more AUTHOR EXTENSIONS a(5) from Marijn Heule, Nov 26 2017 STATUS approved

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Last modified June 23 18:45 EDT 2021. Contains 345402 sequences. (Running on oeis4.)