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.

0, 29, 67, 233, 491, 661, 911, 0, 1747, 2861, 2531, 2857, 7307, 4733, 5791, 7457, 9011, 7309, 14327, 11801, 11047, 14741, 67391, 26737, 16451, 14717, 32779, 41609, 24071, 30661

Offset: 1

Author

Jeremy F. Alm, Nov 04 2017

Keywords

• nonn

Comments

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.

Links

• Jeremy F. Alm, Table of n, a(n) for n = 1..1000
• Jeremy F. Alm, Python program
• Jeremy F. Alm, Directed Ramsey and Anti-Ramsey Algebras and the Flexible Atom Conjecture, arXiv:1901.06781 [math.LO], 2019.
• J. F. Alm and A. Ylvisaker, A fast coset-translation algorithm for computing the cycle structure of Comer relation algebras over Z/pZ, arXiv:1708.04974 [math.CO], 2017.

Crossrefs

Cf. A263308.