%I #21 May 06 2024 23:04:52

%S 1,6,8,12,16,20,24,32,36,40,42,48,54,64,72,80,84,96,108,110,120,126,

%T 128,144,156,160,162,168,192,216,220,240,252,256,272,288,312,320,324,

%U 336,342,378,384,432,440,468,480,486,500,504,506,512,544,550,576,624,640,648,660,672

%N Orders of finite groups for which there is at least one group G such that |Aut(G)| = |G|.

%C Contains A341298 as a subsequence.

%C Every symmetric group S_n of order n! is a member for every n not equal to 2 or 6 since every such S_n is complete.

%F |Out(G)|<=|G| for every entry.

%e a(2) = 6 since the symmetric group of order 6 has 6 automorphisms.

%o (GAP)

%o for n in [1..32] do

%o for i in [1..NrSmallGroups(n)] do

%o if Size(AutomorphismGroup(SmallGroup(n,i))) = n then

%o Print(n,","," ");

%o break;

%o fi;

%o od;

%o od;

%Y Cf. A341298 (orders of complete groups).

%K nonn

%O 1,2

%A _Miles Englezou_, Apr 19 2024