%I A083573
%S A083573 0,0,0,0,0,6,0,10,0,8,0,16,0,10,0
%N A083573 Maximal number of subgroups in a non-Abelian group with n elements.
%C A083573 A group G is non-Abelian iff there are two elements x,y such that xy 
               != yx. Then <x> and <y> are nontrivial subgroups whose order divides 
               the order of G which therefore cannot be prime (neither the square 
               of a prime: there are only two nonisomorphic groups of that order 
               which are both abelian; see A051532 for more). This also implies 
               that a(n) >= 2+2+2 = 6 for all nonzero elements of this sequence 
               and for even n=2m>4 there is the non-Abelian dihedral group D_m with 
               A007503(m)=sigma(m)+tau(m)=A000005(m)+A000203(m), providing a lower 
               bound. - M. F. Hasler (Maximilian.Hasler(AT)gmail.com), Dec 03 2007
%F A083573 a(n) = 0 <=> A060689(n)=0 <=> n is in A051532 ; otherwise a(n) >= 6 and 
               a(2n) >= A007503(n). - M. F. Hasler (Maximilian.Hasler(AT)gmail.com), 
               Dec 03 2007
%e A083573 a(6)=6 because the only non-Abelian group with 6 elements is S_3 with 
               6 subgroups.
%Y A083573 Cf. A018216, A061034.
%Y A083573 Cf. A051532, A060689, A007503.
%Y A083573 Sequence in context: A147709 A153314 A019622 this_sequence A117006 A073764 
               A158897
%Y A083573 Adjacent sequences: A083570 A083571 A083572 this_sequence A083574 A083575 
               A083576
%K A083573 more,nonn
%O A083573 1,6
%A A083573 Victoria A. Sapko (vsapko(AT)canes.gsw.edu), Jun 13 2003