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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.

The term "groupless" was coined by Thomas Chartier.

Andrés Eduardo Caicedo, Thomas A. C. Chartier, Péter Pál Pach, Coloring the n-smooth numbers with n colors, arXiv:1902.00446 [math.NT], 2019.

K. A. Chandler, Groups formed by redefining multiplication, Canad. Math. Bull. Vol. 31 (4), (1988), 419-423.

Th. A. Ch. Chartier, Coloring problems, MS Thesis in Mathematics, Boise State University, December, 2011.

R. Forcade and A. Pollington, What is special about 195? Groups, n-th-power maps and a problem of Graham, in Proceedings of the First Conference of the Canadian Number Theory Association, Banff, 1988, R.A. Mollin, ed., Walter de Gruyter, Berlin, 1990, 147-155.

MathOverflow, Can we color Z^+ with n colors such that a, 2a, …, na all have different colors for all a?

Sequence in context: A183583 A296893 A045073 * A234814 A154938 A234100

Adjacent sequences: A204808 A204809 A204810 * A204812 A204813 A204814

Andres E. Caicedo, Jan 19 2012

approved