A282530 Number of finite FRUTE loops of order n up to isomorphism.

0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 27, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0

Offset

1,8

Comments

For a groupoid Q and x in Q, define the right (left) translation map R_x: Q->Q by yR_x=yx (L_x: Q->Q by yL_x=xy). A loop is a groupoid Q with neutral element 1 in which all translations are bijections in Q. A loop Q is called a FRUTE loop if it satisfies the identity (x.xy)z=(y.xz)x for all x, y, z in Q. The smallest associative non-commutative finite FRUTE loop is of order 8, the quaternion group having 8 elements.

Links

Table of n, a(n) for n=1..63.

T. G. Jaiyeola, A. A. Adeniregun and  M. A. Asiru, Finite FRUTE loops, Journal of Algebra and its Applications, 16:2(2017), 10 pages.

Example

a(8)=2 since there are 2 FRUTE loops of order 8, one of which is the quaternion group of order 8 and a(16)=6 since there are 6 FRUTE loops of order 16.

Crossrefs

Cf. A090750, A281319, A281462, A281554

Sequence in context: A338210 A122698 A002483 * A060478 A088806 A359602

Adjacent sequences: A282527 A282528 A282529 * A282531 A282532 A282533

Keyword

nonn,more

Author

Muniru A Asiru, Feb 17 2017

Status

approved