cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A364129 Order of Aut^3(C_n) = Aut(Aut(Aut(C_n))), where C_n is the cyclic group of order n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 6, 1, 1, 2, 6, 6, 1, 8, 8, 8, 1, 2, 8, 12, 2, 4, 336, 8, 6, 2, 12, 12, 8, 8, 64, 24, 8, 64, 12, 12, 2, 64, 1152, 192, 12, 12, 24, 64, 4, 10, 1152, 12, 8, 768, 64, 16, 2, 128, 336, 24, 12, 12, 1152, 192, 8, 576, 768, 768, 24, 24, 768, 48, 64, 16, 336, 336, 12, 128, 24, 192, 64, 16, 6144
Offset: 1

Views

Author

Jianing Song, Aug 13 2023

Keywords

Examples

			a(24) = 336 since Aut(C_24) = C_2 X C_2 X C_2, Aut^2(C_24) = PSL(2,7) and Aut(Aut(Aut(C_24))) = PGL(2,7).
a(32) = 64 since Aut(C_32) = C_2 X C_8, Aut^2(C_32) = C_2 X D_8 and Aut^3(C_32) = SmallGroup(64,138).
a(40) = 1152 since Aut(C_40) = C_2 X C_2 X C_4, Aut^2(C_40) = SmallGroup(192,1493) and Aut^3(C_40) = C_2 X SmallGroup(576,8654).

Links

Crossrefs

Cf. A000010 (order of Aut(C_n)), A258615 (order of Aut^2(C_n)), A364944 (order of Aut^4(C_n)), A364917 (order of Aut^k(C_n) for all sufficiently large k).

Programs

  • GAP
    A364129 := function(n)
local G, i, L;
G := CyclicGroup(n);
for i in [1..3] do
G := AutomorphismGroup(G);
if i = 3 then return Size(G); 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; end;

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

Views

Author

Jianing Song, Aug 12 2023

Keywords

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).

Examples

			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.

Links

Crossrefs

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

Programs

  • 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;

A364944 Order of Aut^4(C_n) = Aut(Aut(Aut(Aut(C_n)))), where C_n is the cyclic group of order n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 6, 1, 1, 1, 6, 6, 1, 8, 8, 8, 1, 1, 8, 12, 1, 2, 336, 8, 6, 1, 12, 12, 8, 8, 384, 144, 8, 384, 12, 12, 1, 384, 4608, 1152, 12, 12, 144, 384, 2, 4, 4608, 12, 8, 1536, 384, 64, 1, 2359296, 336, 144, 12, 12, 4608, 1152, 8, 13824, 1536, 36864, 144, 24
Offset: 1

Views

Author

Jianing Song, Aug 14 2023

Keywords

Examples

			For n = 69, we have Aut(C_69) = C_2 X C_22, Aut^2(C_69) = C_10 X S_3, Aut^3(C_69) = C_4 X D_12 and Aut^4(C_69) = SmallGroup(32,27) X S_3, so a(69) = |SmallGroup(32,27) X S_3| = 192.
For n = 972, we have Aut(C_972) = C_2 X C_162, Aut^2(C_972) = C_18 X D_12, Aut^3(C_972) = C_6 X S_3 X S_4 and Aut^4(C_972) = C_2 X C_2 X D_12 X S_4, so a(972) = |C_2 X C_2 X D_12 X S_4| = 1152.
For n = 1029, we have Aut(C_1029) = C_2 X C_294, Aut^2(C_1029) = C_42 X D_12, Aut^3(C_1029) = C_6 X D_12 X S_4 and Aut^4(C_1029) = D_12 X S_4 X SmallGroup(96,227), so a(1029) = |D_12 X S_4 X SmallGroup(96,227)| = 27648.
For n = 1944, we have Aut(C_1944) = C_2 X C_2 X C_162, Aut^2(C_1944) = C_2 X C_18 X PSL(2,7), Aut^3(C_1944) = C_6 X S_3 X PGL(2,7) and Aut^4(C_1944) = C_2 X C_2 X D_12 X PGL(2,7), so a(1944) = |C_2 X C_2 X D_12 X PGL(2,7)| = 16128.

Links

Crossrefs

Cf. A000010 (order of Aut(C_n)), A258615 (order of Aut^2(C_n)), A364129 (order of Aut^3(C_n)), A364917 (order of Aut^k(C_n) for all sufficiently large k).

Programs

  • GAP
    A364944 := function(n)
local G, i, L;
G := CyclicGroup(n);
for i in [1..4] do
G := AutomorphismGroup(G);
if i = 4 then return Size(G); 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; end;
# it should be noted that the calculation of Aut^4(C_n) can by extremely lengthy for even small n (for example n = 80)

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

Views

Author

Jianing Song, Aug 18 2023

Keywords

Comments

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.

Examples

			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.

Links

Crossrefs

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

Programs

  • GAP
    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;
Showing 1-4 of 4 results.

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