A045652 - OEIS

%I #55 Oct 19 2024 15:57:32

%S 1,4,13,44,160

%N Schur's numbers (version 2).

%C Largest number such that there is an n-coloring of the integers 1, ..., a(n) such that each color is sum-free, that is, no color contains a triple x + y = z. - _Charles R Greathouse IV_, Jun 11 2013

%C The best known lower bounds for the next terms are due to Fredricksen and Sweet (see links): a(6) >= 536 and a(7) >= 1680. - _Dmitry Kamenetsky_, Oct 23 2019

%C A partition showing that a(7) >= 1696 was demonstrated in 2021, along with some recurrence relationships for lower bounds on a(n). - _Fred Rowley_, Mar 01 2023

%D R. K. Guy, Unsolved Problems in Number Theory, E11 and E12.

%D Marijn J. H. Heule, Schur Number Five, AAAI 2018.

%H Shalom Eliahou, <a href="http://images-archive.math.cnrs.fr/Les-nombres-de-Schur-des-centenaires-pleins-d-avenir.html">Les nombres de Schur, des centenaires pleins d’avenir</a>, Images des Mathématiques, CNRS, 2016 (in French).

%H Harold Fredricksen and Melvin M. Sweet, <a href="https://doi.org/10.37236/1510">Symmetric Sum-Free Partitions and Lower Bounds for Schur Number</a>, The Electronic Journal of Combinatorics, Volume 7, 2000.

%H Solomon W. Golomb and Leonard D. Baumert, <a href="https://doi.org/10.1145/321296.321300">Backtrack Programming</a>, Journal of the ACM 12:4 (1965), pp. 516-524. [Reference corrected by _N. J. A. Sloane_, May 18 2020]

%H Marijn J. H. Heule, <a href="https://arxiv.org/abs/1711.08076">Schur Number Five</a>, arXiv:1711.08076 [cs.LO], 2017.

%H Fred Rowley, <a href="https://arxiv.org/abs/2107.03560">An Improved Lower Bound for S(7) and Some Interesting Templates</a>, arXiv:2107.03560 [math.CO], 2021.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SchurNumber.html">Schur Number</a>

%F a(n) = A030126(n)-1. a(n) <= A003323(n)-2. - _Max Alekseyev_, Jan 12 2008

%e Golomb and Baumert find a(4) = 44 and give this example:

%e A = {1, 3, 5, 15, 17, 19, 26, 28, 40, 42, 44}

%e B = {2, 7, 8, 18, 21, 24, 27, 37, 38, 43}

%e C = {4, 6, 13, 20, 22, 23, 25, 30, 32, 39, 41}

%e D = {9, 10, 11, 12, 14, 16, 29, 31, 33, 34, 35, 36}

%e 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

%e From _Marijn Heule_, Nov 26 2017: (Start)

%e Exoo computed the first certificate showing that a(5) >= 160:

%e 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}

%e B = {2, 3, 8, 14, 19, 20, 24, 25, 36, 46, 47, 51, 62, 73, 88, 99, 110, 114, 115, 125, 136, 137, 141, 142, 147, 153, 158, 159}

%e C = {4, 5, 15, 16, 22, 28, 29, 39, 40, 41, 42, 48, 49, 59, 102, 112, 113, 119, 120, 121, 122, 132, 133, 139, 145, 146, 156, 157}

%e 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 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)

%Y Cf. A030126, A072842.

%K nonn,hard,more,nice

%O 1,2

%A Patric R. J. Östergård (pat(AT)ultra.hut.fi, patric.ostergard(AT)hut.fi)

%E a(5) from _Marijn Heule_, Nov 26 2017

%E Example corrected by _Eckard Specht_, Jul 07 2021