A030126 -id:A030126 - cp's OEIS frontend

A045652 Schur's numbers (version 2).

1, 4, 13, 44, 160

Offset: 1

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

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

			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)

R. K. Guy, Unsolved Problems in Number Theory, E11 and E12.
Marijn J. H. Heule, Schur Number Five, AAAI 2018.

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

Cf. A030126, A072842.

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

a(5) from Marijn Heule, Nov 26 2017

Example corrected by Eckard Specht, Jul 07 2021

A118771 Let a "sum" be a set {x,y,z} of distinct natural numbers such that x+y=z and let N_m={1,2,...m}. a(n) is the smallest s such that there is no partition of N_s into n sum-free parts.

Original entry on oeis.org

3, 9, 24, 67

Offset: 1

R. Reis (rvr(AT)ncc.up.pt), May 22 2006

a(5) >= 190 (see Blanchard et al. at p. 7). - Michel Marcus, Mar 26 2013

a(5) >= 197, a(6) >= 583, a(7) >= 1741, a(8) >= 5202, a(9) >= 15597 (see Ahmed et al. at p. 3). - Stefano Spezia, Aug 25 2023

			For n=1, a(1)=3 as there is no partition of N_3={1,2,3} into 1-sum-free parts. In the same way a(2)=9...

T. Ahmed, L. Boza, M. P. Revuelta, and M. I. Sanz, Exact values and lower bounds on the n-color weak Schur numbers for n=2,3. Ramanujan J (2023). See Table 2 at p. 3.
P. Blanchard, F. Harary, and R. Reis, Partitions into sum-free sets, Integers: electronic journal of combinatorial number theory, 6. 2006.

Cf. A030126, A072842.

A225231 Schur numbers S(3,n).

Original entry on oeis.org

9, 16, 23, 37, 53, 71, 93, 119, 147, 177, 211, 249, 289, 331, 377, 427, 479, 533, 591, 653, 717, 783, 853, 927, 1003, 1081, 1163, 1249, 1337, 1427, 1521, 1619, 1719, 1821, 1927, 2037, 2149, 2263, 2381, 2503, 2627, 2753, 2883, 3017, 3153, 3291, 3433

Offset: 3

Eric M. Schmidt, May 03 2013

a(n) is, by definition, the least positive m such that if {1,...,m} is written as a disjoint union of sets A and B, then either A contains 3 distinct numbers, one the sum of the other two, or B contains n distinct numbers, one the sum of the other n - 1.

Eric M. Schmidt, Table of n, a(n) for n = 3..1000
Tanbir Ahmed, Michael G. Eldredge, Jonathan J. Marler, and Hunter S. Snevily, Strict Schur Numbers, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 13, Paper A22, 2013.
Index entries for linear recurrences with constant coefficients, signature (3,-4,4,-3,1).

Cf. A030126, A045652, A072842.

Join[{9,16},LinearRecurrence[{3,-4,4,-3,1},{23,37,53,71,93},45]] (* Ray Chandler, Feb 13 2014 *)

def A225231(n) : return 9 if n == 3 else 16 if n == 4 else (3*n^2 - 7*n)//2 + [3,3,4,4][n%4]

For n >= 5, a(n) = 3n^2/2 - 7n/2 + c, where c = 3 if n == 0,1 (mod 4), else c = 4.

G.f.: x^3*(3*x^6-7*x^5+3*x^4+4*x^3-11*x^2+11*x-9) / ((x-1)^3*(x^2+1)). - Colin Barker, May 16 2013

A365190 The weak Schur numbers for 2-coloring.

Original entry on oeis.org

9, 24, 52, 101, 166, 253

Offset: 2

Stefano Spezia, Aug 25 2023

a(8) >= 365, a(9) >= 505, a(10) >= 676 (see Table 4 at p. 6 in Ahmed et al.).

T. Ahmed, L. Boza, M. P. Revuelta, and M. I. Sanz, Exact values and lower bounds on the n-color weak Schur numbers for n=2,3. Ramanujan J (2023). See Table 3 at p. 4, Table 4 at p. 6, Sections 4.1 and 4.2 at pp. 10-13.

Cf. A030126, A045652, A072842, A118771, A365191 (similar for 3-coloring).

A365191 The weak Schur numbers for 3-coloring.

Original entry on oeis.org

24, 94, 259

Offset: 2

Stefano Spezia, Aug 25 2023

a(5) >= 593, a(6) >= 1146, a(7) >= 2005 (see Table 6 at p. 8 in Ahmed et al.).

T. Ahmed, L. Boza, M. P. Revuelta, and M. I. Sanz, Exact values and lower bounds on the n-color weak Schur numbers for n=2,3. Ramanujan J (2023). See Table 6 at p. 8 and Sections 4.3 and 4.4 at pp. 13-16.

Cf. A030126, A045652, A072842, A118771, A365190 (similar for 2-coloring).

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A045652 Schur's numbers (version 2).

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A118771 Let a "sum" be a set {x,y,z} of distinct natural numbers such that x+y=z and let N_m={1,2,...m}. a(n) is the smallest s such that there is no partition of N_s into n sum-free parts.

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A225231 Schur numbers S(3,n).

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A365190 The weak Schur numbers for 2-coloring.

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A365191 The weak Schur numbers for 3-coloring.

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