cp's OEIS Frontend

A294615 a(n) is the smallest prime p such that there is a multiplicative subgroup H of Z/pZ, of odd order and of index 2n, such that for any two cosets H1 and H2 of H, H1 + H2 contains all of (Z/pZ)\0, except that H+H contains all of (Z/pZ)\0 except -H. If no such prime exists, a(n) = 0.

%I A294615 #22 Mar 07 2019 18:13:08
%S A294615 0,29,67,233,491,661,911,0,1747,2861,2531,2857,7307,4733,5791,7457,
%T A294615 9011,7309,14327,11801,11047,14741,67391,26737,16451,14717,32779,
%U A294615 41609,24071,30661
%N A294615 a(n) is the smallest prime p such that there is a multiplicative subgroup H of Z/pZ, of odd order and of index 2n, such that for any two cosets H1 and H2 of H, H1 + H2 contains all of (Z/pZ)\0, except that H+H contains all of (Z/pZ)\0 except -H. If no such prime exists, a(n) = 0.
%C A294615 The fact that H is of odd order means H is disjoint from -H. The finite integral relation algebra with n pairs of asymmetric diversity atoms a_i, where the forbidden cycles are of the form (a_i, a_i, a_i^(converse)), is representable over Z/pZ, where p = a(n). These are "directed anti-Ramsey algebras", since "monochromatic intransitive triangles" are forbidden.
%H A294615 Jeremy F. Alm, <a href="/A294615/b294615.txt">Table of n, a(n) for n = 1..1000</a>
%H A294615 Jeremy F. Alm, <a href="/A294615/a294615.py.txt">Python program</a>
%H A294615 Jeremy F. Alm, <a href="https://arxiv.org/abs/1901.06781">Directed Ramsey and Anti-Ramsey Algebras and the Flexible Atom Conjecture</a>, arXiv:1901.06781 [math.LO], 2019.
%H A294615 J. F. Alm and A. Ylvisaker, <a href="https://arxiv.org/abs/1708.04974">A fast coset-translation algorithm for computing the cycle structure of Comer relation algebras over Z/pZ</a>, arXiv:1708.04974 [math.CO], 2017.
%Y A294615 Cf. A263308.
%K A294615 nonn
%O A294615 1,2
%A A294615 _Jeremy F. Alm_, Nov 04 2017