A331921 -id:A331921 - cp's OEIS frontend

A364917 For each n, if the sequence defined by G_0 = C_n, G_k = Aut(G_{k-1}) for k >= 1 stabilizes, then a(n) is the order of G_k for sufficiently large k; otherwise a(n) = 0. Here C_n is the cyclic group of order n.

1, 1, 1, 1, 1, 1, 1, 6, 1, 1, 1, 6, 6, 1, 8, 8, 8, 1, 1, 8, 12, 1, 1, 336, 8, 6, 1, 12, 12, 8, 8

Offset: 1

Jianing Song, Aug 12 2023

It can be checked that a(n) != -1 for the following numbers: 3^k and 2*3^k for all k >= 0; 2*3^k+1 and 2*(2*3^k+1) for all k >= 0 where 2*3^k+1 is a prime; divisors of 16, 20, 26, 30, 34, 46, 50, 58, 62, 74, 94, 118, 134, 146, 168, 172, 178, 196, 218, 258, 264, 294, 346, 394, 456, 458, 648, 686, 694 or 914.
The sequences of iterations are listed as follows (D_{2n} = dihedral group of order 2*n, S_n = symmetric group over set of size n, A_n = alternating group over set of size n):
 - C_{3^k}, C_{2*3^k} -> C_{2*3^(k-1)} -> ... -> C_2 -> C_1, k >= 1 (order = 1);
 - C_{2*3^k+1} or C_{2*(2*3^k+1)} -> C_{2*3^k} -> ... -> C_2 -> C_1, k >= 0, 2*3^k+1 is prime (order = 1);
 - C_47 or C_94 -> C_23 or C_46 -> C_11 or C_22 -> C_5 or C_10 -> C_4 -> C_2 -> C_1 (order = 1);
 - C_13 or C_26 -> C_8 or C_12 -> C_2 X C_2 -> S_3 (order = 6);
 - C_17, C_25, C_31, C_34, C_50 or C_62 -> C_15, C_16, C_20 or C_30 -> C_2 X C_4 -> D_8 (order = 8);
 - C_59 or C_118 -> C_29, C_37, C_43, C_49, C_58, C_74, C_86 or C_98 -> C_21, C_28, C_36 or C_42 -> C_2 X C_6 -> D_12 (order = 12);
 - C_24 -> C_2 X C_2 X C_2 -> PSL(2,7) -> PGL(2,7) (order = 336);
 - C_67 or C_134 -> C_33, C_44 or C_66 -> C_2 X C_10 -> C_4 X S_3 -> C_2 X D_12 -> S_3 X S_4 (order = 144);
 - C_109 or C_218 -> C_57, C_76, C_108 or C_114 -> C_2 X C_18 -> C_6 X S_3 -> C_2 X D_12 -> S_3 X S_4 (order = 144);
 - C_73 or C_146 -> C_56, C_72 or C_84 -> C_2 X C_2 X C_6 -> C_2 X PSL(2,7) -> PGL(2,7) (order = 336);
 - C_89 or C_178 -> C_88 or C_132 -> C_2 X C_2 X C_10 -> C_4 X PSL(2,7) -> C_2 X PGL(2,7) (order = 672);
 - C_347 or C_694 -> C_173, C_197, C_343, C_346, C_394 or C_686 -> C_129, C_147, C_172, C_196, C_258 or C_294 -> C_2 X C_42 -> C_6 X D_12 -> C_2 X D_12 X S_4 -> S_3 X S_4 X SmallGroup(96,227) -> S_3 X S_4 X SmallGroup(576,8654) -> S_3 X S_4 X SmallGroup(1152,157849) (order = 165888);
 - C_229 or C_458 -> C_152, C_216 or C_228 -> C_2 X C_2 X C_18 -> C_6 X PSL(2,7) -> C_2 X PGL(2,7).(order = 672);
 - C_168 -> C_2 X C_2 X C_2 X C_6 -> C_2 X A_8 -> S_8 (order = 40320);
 - C_264 -> C_2 X C_2 X C_2 X C_10 -> C_4 X A_8 -> C_2 X S_8 (order = 80640);
 - C_457 or C_914 -> C_456 -> C_2 X C_2 X C_2 X C_18 -> C_6 X A_8 -> C_2 X S_8 (order = 80640);
 - C_324 -> C_2 X C_54 -> C_18 X S_3 -> C_6 X D_12 -> C_2 X D_12 X S_4 -> S_3 X S_4 X SmallGroup(96,227) -> S_3 X S_4 X SmallGroup(576,8654) -> S_3 X S_4 X SmallGroup(1152,157849) (order = 165888);
 - C_648 -> C_2 X C_2 X C_54 -> C_18 X PSL(2,7) -> C_6 X PGL(2,7) -> C_2 X C_2 X PGL(2,7) -> S_4 X PGL(2,7) (order = 8064).
The following two sequences are conjectured to be correct and to stabilize at the last term:
 - C_344, C_392, C_516 or C_588 -> C_2 X C_2 X C_42 -> C_2 X C_6 X PSL(2,7) - > D_12 X PGL(2,7) -> C_2 X D_12 X PGL(2,7) -> S_3 X PGL(2,7) X SmallGroup(96,227) -> S_3 X PGL(2,7) X SmallGroup(576,8654)? -> S_3 X PGL(2,7) X SmallGroup(1152,157849)? (order = 2322432);
 - C_1033 or C_2066 -> C_1032 or C_1176 -> C_2 X C_2 X C_2 X C_42 -> C_2 X C_6 X A_8 - > D_12 X S_8 -> C_2 X D_12 X S_8? -> S_3 X S_8 X SmallGroup(96,227)? -> S_3 X S_8 X SmallGroup(576,8654)? -> S_3 X S_8 X SmallGroup(1152,157849)? (order = 278691840).
Note that a(35) = a(32) (although the value of each is unknown), since Aut(Aut(C_32)) = Aut(Aut(C_35)) = C_2 X D_8.

			For n = 24, we have G_0 = C_24, G_1 = C_2 X C_2 X C_2, G_2 = PSL(2,7) and G_k = PGL(2,7) for all k >= 3, hence a(24) = |PGL(2,7)| = 336.

G. Muller, Does Aut(Aut(...Aut(G)...)) stabilize?, MathOverflow (2009).
S. Palcoux, On the iterated automorphism groups of the cyclic groups, MathOverflow (2020).

Cf. A331921, A117729 (indices of 1).

A117729 Orders k of cyclic groups C_k such that the map "G -> Automorphism group of G" eventually reaches the trivial group when started at C_k.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 14, 18, 19, 22, 23, 27, 38, 46, 47, 54, 81, 94, 162, 163, 243, 326, 486, 487, 729, 974, 1458, 1459, 2187, 2918, 4374, 6561, 13122, 19683, 39366, 39367, 59049, 78734, 118098, 177147, 354294, 531441, 1062882, 1594323, 3188646, 4782969

Offset: 1

N. J. A. Sloane, based on a communication from J. H. Conway, Apr 14 2006

nonn

If the map "G -> Automorphism group of G" eventually reaches the trivial group, then the initial group IS a cyclic group.
From Jianing Song, Oct 12 2019: (Start)
These are numbers k such that every step of the iteration results in a cyclic group, i.e., numbers k such that k, phi(k), phi(phi(k)), phi(phi(phi(k))), ... (or equivalently, k, A258615(k), A258615(A258615(k)), ...) are all in A033948, phi = A000010.
Number of iterations to reach the trivial group:
  k = 1: 0;
  k = 2: 1;
  k = 4: 2;
  k = 5, 10: 3;
  k = 11, 22: 4;
  k = 23, 46: 5;
  k = 47, 94: 6;
  k = 3^i, 2*3^i, i > 0: i+1;
  k = 2*3^i+1, 2*(2*3^i+1), i > 0, 2*3^i+1 is prime: i+2. (End)
From Peter Schorn, Apr 06 2021: (Start)
Since the values of a(n) have a simple formula it is easy to confirm by direct calculation for all cases that A003434(a(n)) = A185816(a(n)), i.e., the number of iterations to reach 1 via the Euler phi function is the same as the number of iterations to reach 1 via the Carmichael lambda function.
A computer search up to n = 10^8 also confirms the conjecture that if A003434(n) = A185816(n) then n is a term of A117729.
(End)

Cf. A033948, A000010, A258615, A331921, A003434, A185816.

t1:={ 4, 5, 10, 11, 22, 23, 46, 47, 94}; for i from 0 to 30 do t1:={op(t1),3^i, 2*3^i}; if isprime(2*3^i+1) then t1:={op(t1), 2*3^i+1, 2*(2*3^i+1)}; fi; od: convert(t1,list); sort(%);

ok(k)={my(f=1, t); while(f&&k>1, f=if(k%2, isprimepower(k), k==2 || k==4 || (isprimepower(k/2, &t) && t>2)); k=eulerphi(k)); f}
{ for(n=1, 10^9, if(ok(n), print1(n, ", "))) } \\ Andrew Howroyd, Oct 12 2019

Consists of the following numbers:
3^i and 2*3^i for all i >= 0;
if 2*3^i+1 is a prime, then also 2*3^i+1 and 2(2*3^i+1);
the exceptional entries 4, 5, 10, 11, 22, 23, 46, 47 and 94.

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.

Original entry on oeis.org

32, 16, 16, 64, 384, 1536, 6144

Offset: 0

Jianing Song, Aug 12 2023

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).

			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.

G. Muller, Does Aut(Aut(...Aut(G)...)) stabilize?, MathOverflow (2009).
S. Palcoux, On the iterated automorphism groups of the cyclic groups, MathOverflow (2020).

Cf. A365051 ({Aut^n(C_40)}), A364917, A331921.

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;

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.

Original entry on oeis.org

40, 16, 192, 1152, 4608, 18432

Offset: 0

Jianing Song, Aug 18 2023

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.

			Aut(C_40) = C_2 X C_2 X C_4, so a(1) = 16;
Aut^2(C_40) = SmallGroup(192,1493), so a(2) = 192;
Aut^3(C_40) = SmallGroup(192,1493), so a(3) = 1152.

G. Muller, Does Aut(Aut(...Aut(G)...)) stabilize?, MathOverflow (2009).
S. Palcoux, On the iterated automorphism groups of the cyclic groups, MathOverflow (2020).

Cf. A364904 ({Aut^n(C_32)}), A364917, A331921.

A365051 := 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;

cp's OEIS Frontend

A364917 For each n, if the sequence defined by G_0 = C_n, G_k = Aut(G_{k-1}) for k >= 1 stabilizes, then a(n) is the order of G_k for sufficiently large k; otherwise a(n) = 0. Here C_n is the cyclic group of order n.

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A117729 Orders k of cyclic groups C_k such that the map "G -> Automorphism group of G" eventually reaches the trivial group when started at C_k.

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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.

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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.

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