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URL: https://oeis.org/A364904

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A364904
a(n) = |Aut^n(C_32)|: order of the group obtained by applying G -> Aut(G) n times to the cyclic group of order 32.
2
32, 16, 16, 64, 384, 1536, 6144
OFFSET
0,1
COMMENTS
Also a(n) = |Aut^n(C_35)| for n >= 2, since Aut(Aut(C_32)) = Aut(Aut(C_35)) = C_2 X D_8.
The sequence {Aut^n(C_m):n>=0} is well-known for m <= 31. It is conjectured that |Aut^n(C_32)| tends to infinity as n goes to infinity.
This sequence appears in the table shown in the Math Overflow question "On the iterated automorphism groups of the cyclic groups" (see the Links section below).
EXAMPLE
Aut(C_32) = C_2 X C_8, so a(1) = 16;
Aut^2(C_32) = C_2 X D_8, so a(2) = 16;
Aut^3(C_32) = SmallGroup(64,138), so a(3) = 64;
Aut^4(C_32) = SmallGroup(384,17948), so a(4) = 384.
PROG
(GAP) A364904 := function(n)
local G, i, L;
G := CyclicGroup(32);
for i in [1..n] do
G := AutomorphismGroup(G);
if i = n then return break; fi;
L := DirectFactorsOfGroup(G);
if List(L, x->IdGroupsAvailable(Size(x))) = List(L, x->true) then
L := List(L, x->IdGroup(x));
G := DirectProduct(List(L, x->SmallGroup(x))); # It's more efficient to operate on abstract groups when the abstract structure is available
fi; od;
return Size(G);
end;
CROSSREFS
Cf. A365051 ({Aut^n(C_40)}), A364917, A331921.
Sequence in context: A070621 A234967 A033352 * A140387 A023094 A087502
KEYWORD
nonn,hard,more
AUTHOR
Jianing Song, Aug 12 2023
STATUS
approved