%I A147848
%S A147848 1,2,2,4,2,4,2,6,2,4,2,8,2,4,4,6,2,4,2,8,4,4,2,12,2,4,2,8,2,8,2,6,4,4,
               4,
%T A147848 8,2,4,4,12,2,8,2,8,4,4,2,12,2,4,4,8,2,4,4,12,4,4,2,16
%N A147848 Number (up to isomorphism) of groups of order 2n that have Z/nZ as a 
               subgroup (that is, that have an element of order n).
%C A147848 This sequence is related to A060594 : in fact, for every square root 
               of unity modulo n, there are either one or two such groups of order 
               2n.
%F A147848 a(2) = 2; a(4) = 4; a(2^k) = 6 for k >= 3.
%F A147848 a(p^k) = 2 for any odd prime number p and k >= 1.
%F A147848 For other values of n, you can find a(n) by using the fact that the sequence 
               is multiplicative.
%e A147848 Two such groups that always exist are the cyclic group Z/(2n)Z and the 
               dihedral group Dih_n. If n is prime, these are the only such groups, 
               so the n-th term equals 2.
%e A147848 For even values of n, we also have the direct product Z/nZ x Z/2Z and 
               the dicyclic group Dic_n. If n = 2p with p prime, there are no other 
               groups, so the n-th term equals 4.
%Y A147848 Cf. A060594.
%Y A147848 Sequence in context: A001223 A118776 A092520 this_sequence A129089 A124315 
               A101113
%Y A147848 Adjacent sequences: A147845 A147846 A147847 this_sequence A147849 A147850 
               A147851
%K A147848 easy,nice,nonn,mult
%O A147848 1,2
%A A147848 Ilia Smilga (ilia.smilga(AT)ens.fr), Nov 15 2008
%E A147848 Extended comments, references and confirmed "mult" keyword. - Ilia Smilga 
               (ilia.smilga(AT)ens.fr), Nov 17 2008