A057771 -id:A057771 - cp's OEIS frontend

A205645 Number of nilpotent loops of order 2*prime(n) up to isotopism.

3, 64, 3658004, 1023090941561683953759580, 2684673506279593406254437209960379084, 382103603974564085117495134243710834769544696954218618882023686506660

Offset: 2

Jonathan Vos Post, Jan 29 2012

nonn

Table 5, p. 20, of Clavier. oddprime(n) = A065091(n) = A000040(n-1).

In version 2 of Clavier's preprint as well as in the published paper, the numbers have been replaced by numbers 1 smaller: 2, 63, 3658003... Theorem 6.10, which gives a formula, has also been changed. The arXiv source files include a file with all terms for which prime(n) < 100. That file agrees with this sequence rather than with the sequence in the updated version of the preprint (or in the published paper). - Andrei Zabolotskii, May 06 2025

			a(4) = 3658004 because prime(4) = 7 and there are 3658004 nilpotent loops of order 2*7 = 14.

L. Clavier, About the autotopisms of abelian groups, 2012.
D. Daly and P. Vojtěchovský, Enumeration of nilpotent loops via cohomology, J. Algebra, 322(11):4080-4098, 2009.
H.O. Pflugfelder, Quasigroups and Loops: Introduction, 1990.
J. D. Phillips and P. Vojtěchovský, The varieties of loops of bolmoufang type, Algebra Universalis, 54(3):259-271, 2005.

Lucien Clavier, Enumeration of nilpotent loops up to isotopy, arXiv:1201.5659v1 [math.GR], Jan 26, 2012.
Lucien Clavier, Enumeration of nilpotent loops up to isotopy, Commentationes Mathematicae Universitatis Carolinae, 53 (2012), 159-177.

Cf. A000040, A057771, A065091.

A318986 Number of loops of order n that are not groups.

Original entry on oeis.org

0, 0, 0, 0, 0, 5, 107, 23745, 106228844, 9365022303538, 20890436195945769615, 1478157455158044452849321015

Offset: 0

Steve Szabo, Sep 06 2018

A loop is a quasigroup with an identity element. - Muniru A Asiru, Dec 13 2018

A. Hulpke, Petteri Kaski and Patric R. J. Östergård, The number of Latin squares of order 11, Math. Comp. 80 (2011) 1197-1219.
Brendan D. McKay, A. Meynert and W. Myrvold, Small Latin Squares, Quasigroups and Loops, J. Combin. Designs 15 (2007), no. 2, 98-119.
Wikipedia, Quasigroup
Index entries for sequences related to quasigroups

Cf. A057771, A000001.

a(n) = A057771(n) - A000001(n).

a(7)-a(11) from Muniru A Asiru, Dec 07 2018

a(0) prepended by Jianing Song, Oct 26 2019

A328746 Number of loops of order n, considered to be equivalent when they are isomorphic or anti-isomorphic (by reversal of the operator).

Original entry on oeis.org

0, 1, 1, 1, 2, 5, 72, 12151, 53146457

Offset: 0

Jianing Song, Oct 26 2019

For the number of group-like algebraic structures of order n, see:
Semigroups: A027851 or A001423 (commutative: A001426);
Monoids: A058129 or A058133 (commutative: A058131);
Quasigroups: A057991 or A058171 (commutative: A057992);
Loops: A057771 or this sequence (commutative: A089925);
Groups: A000001 (commutative: A000688);
Rings: A027623 or A038036 (commutative: A037289);
Rings with unity: A037291;
Fields: A069513.

a(n) = (A057771(n)+A057996(n))/2.

cp's OEIS Frontend

A205645 Number of nilpotent loops of order 2*prime(n) up to isotopism.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

A318986 Number of loops of order n that are not groups.

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

Extensions

A328746 Number of loops of order n, considered to be equivalent when they are isomorphic or anti-isomorphic (by reversal of the operator).

Views

Author

Keywords

Crossrefs

Formula