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A365051 a(n) = |Aut^n(C_40)|: order of the group obtained by applying G -> Aut(G) n times to the cyclic group of order 40.

%I A365051 #10 Aug 19 2023 16:12:37
%S A365051 40,16,192,1152,4608,18432
%N A365051 a(n) = |Aut^n(C_40)|: order of the group obtained by applying G -> Aut(G) n times to the cyclic group of order 40.
%C A365051 m = 40 is the next case after m = 32 where the sequence {Aut^n(C_m):n>=0} is not known to stabilize after some n. See A364904.
%H A365051 G. Muller, <a href="https://mathoverflow.net/q/5635/34538">Does Aut(Aut(...Aut(G)...)) stabilize?</a>, MathOverflow (2009).
%H A365051 S. Palcoux, <a href="https://mathoverflow.net/q/351593/34538">On the iterated automorphism groups of the cyclic groups</a>, MathOverflow (2020).
%e A365051 Aut(C_40) = C_2 X C_2 X C_4, so a(1) = 16;
%e A365051 Aut^2(C_40) = SmallGroup(192,1493), so a(2) = 192;
%e A365051 Aut^3(C_40) = SmallGroup(192,1493), so a(3) = 1152.
%o A365051 (GAP) A365051 := function(n)
%o A365051 local G, i, L;
%o A365051 G := CyclicGroup(32);
%o A365051 for i in [1..n] do
%o A365051 G := AutomorphismGroup(G);
%o A365051 if i = n then return break; fi;
%o A365051 L := DirectFactorsOfGroup(G);
%o A365051 if List(L, x->IdGroupsAvailable(Size(x))) = List(L, x->true) then
%o A365051 L := List(L, x->IdGroup(x));
%o A365051 G := DirectProduct(List(L, x->SmallGroup(x))); # It's more efficient to operate on abstract groups when the abstract structure is available
%o A365051 fi; od;
%o A365051 return Size(G);
%o A365051 end;
%Y A365051 Cf. A364904 ({Aut^n(C_32)}), A364917, A331921.
%K A365051 nonn,hard,more
%O A365051 0,1
%A A365051 _Jianing Song_, Aug 18 2023