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A045652 Schur's numbers (version 2). 5
1, 4, 13, 44, 160 (list; graph; refs; listen; history; text; internal format)
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

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.

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

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

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

Baumert & Golomb 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, 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. 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 Ostergard (pat(AT)ultra.hut.fi, patric.ostergard(AT)hut.fi)

EXTENSIONS

a(5) from Marijn Heule, Nov 26 2017

STATUS

approved

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Last modified March 26 05:43 EDT 2019. Contains 321481 sequences. (Running on oeis4.)