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A281319 Number of left Bol loops (including Moufang loops) of order n which are not groups.

%I A281319 #26 Mar 29 2023 07:34:09
%S A281319 0,0,0,0,0,0,0,6,0,0,0,3,0,0,2,2038,0,2,0,3,2,0,0
%N A281319 Number of left Bol loops (including Moufang loops) of order n which are not groups.
%C A281319 A loop is a set L with binary operation (denoted simply by juxtaposition) such that for each a in L, the left (right) multiplication map L_a:=L->L, x->xa (R_a: L->L, x->ax) is bijective and L has a two-sided identity 1.  A loop is left Bol if it satisfies the left Bol identity (x.yx)z=x(y.xz) for all x,y,z in L. A loop is Moufang if it is both left Bol and right Bol.
%D A281319 E. G. Goodaire and S. May, Bol loops of order less than 32, Dept of Math and Statistics, Memorial University of Newfoundland, Canada, 1995.
%H A281319 R. P. Burn, <a href="https://doi.org/10.1017/S0305004100063556">Corrigenda: Finite Bol loops: III</a>, Math. Proc. Camb. Phil. Soc. (1985), 98, 381.
%H A281319 Michael K. Kinyon, Gábor P. Nagy and Petr Vojtěchovský, <a href="https://doi.org/10.1016/j.jalgebra.2016.11.023">Bol loops and Bruck loops of order pq</a>, Journal of Algebra, Volume 473, 2017, Pages 481-512.
%H A281319 Eric Moorhouse, <a href="https://ericmoorhouse.org/pub/bol/">Bol loops of small orders</a>
%H A281319 B. L. Sharma, <a href="https://dml.cz/dmlcz/142527">Classification of Bol loops of order 18</a>, Acta Universitatis Carolinae. Mathematica et Physica 025.1 (1984): 37-44.
%H A281319 B. L. Sharma and A. R. T. Solarin, <a href="http://doi.org/10.1080/00927878808823560">On classification of Bol loops of order 3p (p>3)</a>, Comm. in Algebra 16:1(1988), 37-55.
%e A281319 a(8)=6 since there are 6 left Bol loops of order 8 and a(12)=3 since there are 3 left Bol loops of order 12 one of which is the smallest Moufang loop.
%Y A281319 Cf. A090750.
%K A281319 nonn,more
%O A281319 1,8
%A A281319 _Muniru A Asiru_, Jan 20 2017
%E A281319 a(18) changed to 2 by _N. J. A. Sloane_, Feb 02 2023 at the suggestion of Kurosh Mavaddat Nezhaad, who said in an email that the number of Bol loops of order 18, and generally of order 2p^2 up to isomorphism, is exactly 2. See Sharma (1984) or Burn (1985).