%I A140185
%S A140185 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,
%T A140185 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,
%U A140185 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
%N A140185 Number of connected groupoids (categories all of whose morphisms are 
               invertible) with n morphisms.
%C A140185 a(n) >= A000001(n) (number of groups of order n), with equality if and 
               only if n is square-free (sequence A005117).
%H A140185 B. Jubin, <a href="http://math.berkeley.edu/~jubin/oeis.html">Sequences 
               contributed to the OEIS</a>.
%F A140185 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 non-zero values are b(0,0) = 1 and b(m.k^2,k) = A000001(m) 
               with m,k>0.
%Y A140185 Cf. A140186, A140187, A140188, A140189.
%Y A140185 Sequence in context: A010123 A039620 A008296 this_sequence A106790 A078897 
               A011086
%Y A140185 Adjacent sequences: A140182 A140183 A140184 this_sequence A140186 A140187 
               A140188
%K A140185 nonn
%O A140185 0,5
%A A140185 Benoit Jubin (benoit_jubin(AT)yahoo.fr), May 12 2008