A140185 Number of connected groupoids (categories all of whose morphisms are invertible) with n morphisms.

1, 1, 1, 1, 3, 1, 2, 1, 6, 3, 2, 1, 6, 1, 2, 1, 17, 1, 6, 1, 6, 2, 2, 1, 17, 3, 2, 6, 5, 1, 4, 1, 57, 1, 2, 1, 19, 1, 2, 2, 16, 1, 6, 1, 5, 3, 2, 1, 58, 3, 6, 1, 6, 1, 17, 2, 15, 2, 2, 1, 14, 1, 2, 5, 284, 1, 4, 1, 6, 1, 4, 1, 61, 1, 2, 4, 5, 1, 6, 1, 58, 18, 2, 1, 17, 1, 2, 1, 14, 1, 12, 1, 5, 2

Offset: 0

Author

Benoit Jubin, May 12 2008

Keywords

• nonn

Comments

a(n) >= A000001(n) (number of groups of order n), with equality if and only if n is squarefree (sequence A005117).

Crossrefs

Cf. A140186, A140187, A140188, A140189.

Formula

a(0) = 1 and if n>0, a(n) = sum(A000001(n/(k^2)),k^2|n). Indeed, if b(n,k) is the number of connected groupoids with n morphisms and k objects, then the only nonzero values are b(0,0) = 1 and b(m.k^2,k) = A000001(m) with m,k>0.