A045652 Schur's numbers (version 2).

1, 4, 13, 44, 160

Offset

1,2

Comments

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

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

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

References

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

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

Links

Table of n, a(n) for n=1..5.

Shalom Eliahou, Les nombres de Schur, des centenaires pleins d’avenir, Images des Mathématiques, CNRS, 2016 (in French).

Harold Fredricksen and Melvin M. Sweet, Symmetric Sum-Free Partitions and Lower Bounds for Schur Number, The Electronic Journal of Combinatorics, Volume 7, 2000.

Solomon W. Golomb and Leonard D. Baumert, Backtrack Programming, Journal of the ACM 12:4 (1965), pp. 516-524. [Reference corrected by N. J. A. Sloane, May 18 2020]

Marijn J. H. Heule, Schur Number Five, arXiv:1711.08076 [cs.LO], 2017.

Fred Rowley, An Improved Lower Bound for S(7) and Some Interesting Templates, arXiv:2107.03560 [math.CO], 2021.

Eric Weisstein's World of Mathematics, Schur Number

Formula

a(n) = A030126(n)-1. a(n) <= A003323(n)-2. - Max Alekseyev, Jan 12 2008

Example

Golomb and Baumert find a(4) = 44 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}
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, Nov 26 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, 46, 47, 51, 62, 73, 88, 99, 110, 114, 115, 125, 136, 137, 141, 142, 147, 153, 158, 159}
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}
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. A030126, A072842.

Sequence in context: A283110 A149428 A149429 * A149430 A124463 A081197

Adjacent sequences: A045649 A045650 A045651 * A045653 A045654 A045655

Keyword

nonn,hard,more,nice

Author

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

Extensions

a(5) from Marijn Heule, Nov 26 2017

Example corrected by Eckard Specht, Jul 07 2021

Status

approved