A390164 Number of nonisomorphic magmas with n elements satisfying the equation (xy)z = xy.

1, 1, 2, 6, 89, 7922, 11487694, 189629342061, 48995364862569007, 448354248418525120488125, 83174376474244578069383415417540, 345724085634340317989806473617449723287905, 92172779273993943383908703087314593532707007587739759

Offset

0,3

Comments

Every element e in the image of the magma has the property that e*x = e for any x. There are no other constraints.

Links

Andrew Howroyd, Table of n, a(n) for n = 0..36

Example

The Cayley tables of the a(3) = 6 magmas are:
   1 1 1   1 1 1   1 1 1   1 1 1   1 1 1   1 1 1
   1 1 1   2 2 2   2 2 2   2 2 2   2 2 2   2 2 2
   1 1 1   1 1 1   1 1 2   1 2 1   2 1 1   3 3 3

Prog

(PARI)
permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
C(d, r) = {sum(i=1, #r, my(t=r[i]); if(d%t==0, t))}
E(u, v, r) = {prod(i=1, #u, prod(j=1, #v, my(g=gcd(u[i], v[j])); C(u[i]*v[j]/g, r)^g ))}
a(n) = {if(n==0, 1, sum(k=1, n, my(s=0); forpart(r=k, forpart(p=n-k, s+=permcount(r) * permcount(p) * E(p, p, r) * E(p, r, r) )); s/(k!*(n-k)!)))}

Crossrefs

Cf. A001329 (magmas), A362384, A362385, A390603 (labeled case).

Sequence in context: A364794 A334779 A305598 * A179214 A128265 A002432

Adjacent sequences: A390161 A390162 A390163 * A390165 A390166 A390167

Keyword

nonn

Author

Andrew Howroyd, Nov 18 2025

Status

approved