%I
%S 195,205,208,211,212,214,217,218,220,227,229,235,242,244,246,247,248,
%T 252,253,255,257,258,259,263,264,265,266,267,269,271,274,275,279,283,
%U 286,287,289,290,291,294,295,297,298
%N Groupless numbers.
%C Let [n]={1,..,n}. A number n is groupless iff there is no binary operation . on [n] such that G=([n],.) is a group, and . extends the partial graph of multiplication on [n], i.e., whenever i,j and their usual product i*j are in [n], then i.j=i*j. If n is not groupless, a witness G is sometimes called an FLP group.
%C The term "groupless" was coined by Thomas Chartier.
%H Andrés Eduardo Caicedo, Thomas A. C. Chartier, Péter Pál Pach, <a href="https://arxiv.org/abs/1902.00446">Coloring the nsmooth numbers with n colors</a>, arXiv:1902.00446 [math.NT], 2019.
%H K. A. Chandler, <a href="http://dx.doi.org/10.4153/CMB19880615">Groups formed by redefining multiplication, Canad. Math. Bull. Vol. 31 (4), (1988), 419423.
%H Th. A. Ch. Chartier, <a href="http://scholarworks.boisestate.edu/td/231/">Coloring problems</a>, MS Thesis in Mathematics, Boise State University, December, 2011.
%H R. Forcade and A. Pollington, <a href="https://doi.org/10.1515/9783110848632015">What is special about 195? Groups, nthpower maps and a problem of Graham</a>, in Proceedings of the First Conference of the Canadian Number Theory Association, Banff, 1988, R.A. Mollin, ed., Walter de Gruyter, Berlin, 1990, 147155.
%H MathOverflow, <a href="http://mathoverflow.net/questions/26358">Can we color Z^+ with n colors such that a, 2a, …, na all have different colors for all a?</a>
%K nonn
%O 1,1
%A _Andres E. Caicedo_, Jan 19 2012
