This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A362137 #27 Jul 24 2024 23:56:55 %S A362137 1,3,7,19,67,331,1163 %N A362137 Smallest size of an n-paradoxical tournament built as a directed Paley graph. %C A362137 An n-paradoxical tournament consists of a complete oriented 1-graph (each pair of vertices are connected by exactly one directed edge) in which all possible groups of n vertices have a common predecessor. %C A362137 A Paley graph is constructed from the members of a finite field F by connecting pairs of elements that differ by a quadratic residue. %C A362137 In an n-paradoxical tournament built as a directed Paley graph, a vertex x is the predecessor of a vertex y if and only if y-x is a quadratic residue of F. %C A362137 a(0)=1, a(1)=3, a(2)=7 and a(3)=19 are proved to be the smallest sizes of an n-paradoxical tournament. The following a(4)=67, a(5)=331 and a(6)=1163 are only the smallest sizes of the known solutions for an n-paradoxical tournament but they are the smallest sizes of an n-paradoxical tournament built as a directed Paley graph. %C A362137 All known smallest sizes of an n-paradoxical tournament are primes congruent to 3 mod 4. %C A362137 No reasonable values of a(n) for n > 6 are known. %C A362137 Lower and upper bounds are given in the papers given in the references section. %H A362137 P. Erdős, <a href="https://www.jstor.org/stable/3613396">On a Problem in Graph Theory</a>, The Mathematical Gazette, 47.361 (1963), 220-223. %H A362137 R. L. Graham and J. H. S. Spencer, <a href="https://doi.org/10.4153/CMB-1971-007-1">A Constructive Solution to a Tournament Problem</a>, Canadian Mathematical Bulletin 14.1, (1971), 45-48. %H A362137 K. B. Reid and A. A. McRae and S.M. Hedetniemi and S. T. Hedetniemi, <a href="https://ajc.maths.uq.edu.au/pdf/29/ajc_v29_p157.pdf">Domination and irredundance in tournaments</a>, Australas. J Comb., 29 (2004), 157-172. %H A362137 E. Szekeres and G. Szekeres, <a href="https://www.jstor.org/stable/3612854">On a Problem of Schütte and Erdős</a>, The Mathematical Gazette 49.369 (1965), 290-293. %e A362137 For n=1, a(1)=3 vertices, each one being the predecessor of exactly one of the other two. %e A362137 For n=2, a(2)=7 vertices named 0,1,2,3,4,5,6, each vertex x being the predecessor of vertices x+1, x+2, x+4 mod 7. %e A362137 For n=3, a(3)=19 vertices named 0,1,2,...,18, each vertex x being the predecessor of vertices x+1, x+4, x+5, x+6, x+7, x+9, x+11, x+16, x+17 mod 19. %K A362137 nonn,hard,more %O A362137 0,2 %A A362137 _Julien Rouyer_, Jun 12 2023 %E A362137 a(6) corrected by _Nicholas Stefan Georgescu_, Jul 03 2024