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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

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

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

Eric W. Weisstein

EXTENSIONS

a(5) from Marijn Heule, Nov 26 2017

STATUS

approved

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Last modified June 25 12:55 EDT 2019. Contains 324352 sequences. (Running on oeis4.)