A185816 -id:A185816 - cp's OEIS frontend

A027763 Smallest k such that 2^^n is not congruent to 2^^(n-1) mod k, where 2^^n denotes the power tower 2^2^...^2 (in which 2 appears n times).

2, 3, 5, 11, 23, 47, 283, 719, 1439, 2879, 34549, 138197, 531441, 1594323, 4782969, 14348907, 43046721, 86093443, 344373773, 688747547, 3486784401

Offset: 1

R. K. Guy

This sequence shares many terms with A056637, the least prime of class n-. Note that 3^(n-1) is an upper bound for each term and the upper bound is reached for n=13 and n=14. Are all subsequent terms 3^(n-1)? The Mathematica code uses the TowerMod function in the CNT package, which is described in the book by Bressoud and Wagon. - T. D. Noe, Mar 13 2009
For n=15, n=16, and n=17, the terms are also of the form 3^(n-1), but for n=18 and n=19, the terms are prime. - Wayne VanWeerthuizen, Aug 26 2014
A185816(a(n)) = n. - Reinhard Zumkeller, Sep 02 2014
Prime terms seen up to n=20 are in eleven instances of the form j*a(n-1)+1, for j=2, 4, 6, or 12. Note, though, that a(2)=5 and a(8)=719 are exceptions to this pattern. - Wayne VanWeerthuizen, Sep 06 2014

			2^^2=2^2=4 and 2^^3=2^2^2=16. We find 4 = 16 (mod k) until k=5. So a(3)=5. - _T. D. Noe_, Mar 13 2009

David Bressoud and Stan Wagon, A Course in Computational Number Theory, Key College Pub., 2000, p. 96.
Stan Wagon, posting to Problem of the Week mailing list, Dec 15 1997.

D. Bressoud, CNT.m Computational Number Theory Mathematica package.
Stan Wagon, Putnam Problem Notes
Eric W. Weisstein, MathWorld: Power Tower

Cf. A056637, A173927, A185816, A246491.

Needs["CNT`"]; k=1; Table[While[TowerMod[2,n,k]==TowerMod[2,n-1,k], k++ ]; k, {n,10}] (* T. D. Noe, Mar 13 2009 *)

Improved the name and changed the offset because I just prepended a term. - T. D. Noe, Mar 13 2009
Corrected and extended by T. D. Noe, Mar 13 2009
Terms a(15)-a(19) from Wayne VanWeerthuizen, Aug 26 2014
Terms a(20)-a(21) from Wayne VanWeerthuizen, Sep 06 2014

A173927 Smallest integer k such that the number of iterations of Carmichael lambda function (A002322) needed to reach 1 starting at k (k is counted) is n.

Original entry on oeis.org

1, 2, 3, 5, 11, 23, 47, 283, 719, 1439, 2879, 34549, 138197, 531441, 1594323, 4782969, 14348907, 43046721, 86093443, 258280327, 688747547

Offset: 1

Michel Lagneau, Nov 26 2010

Smallest number k such that the trajectory of k under iteration of Carmichael lambda function contains exactly n distinct numbers (including k and the fixed point).
The first 13 terms are 1 or a prime. The next five terms are powers of 3. Then another prime. What explains this behavior? - T. D. Noe, Mar 23 2012
A185816(a(n)) = n - 1. - Reinhard Zumkeller, Sep 02 2014
If a(n) (n > 1) is either a prime or a power of 3, then a(n) is also the smallest integer k such that the number of iterations of Euler's totient function (A000010) needed to reach 1 starting at k (k is counted) is n. - Jianing Song, Jul 10 2019

			for n=5, a(5)=11 gives a chain of length 5 because the trajectory is 11 -> 10 -> 4 -> 2 -> 1.

Nick Harland, The number of iterates of the Carmichael lambda function required to reach 1, arXiv:1203.4791v1 [math.NT], Mar 21 2012.

Cf. A002322, A027763, A056637.

Cf. A185816 (number of iterations of Carmichael lambda function needed to reach 1), A003434 (number of iterations of Euler's totient function needed to reach 1).

import Data.List (elemIndex); import Data.Maybe (fromJust)
a173927 = (+ 1) . fromJust . (`elemIndex` map (+ 1) a185816_list)
-- Reinhard Zumkeller, Sep 02 2014

f[n_] := Length@ NestWhileList[ CarmichaelLambda, n, Unequal, 2] - 1; t = Table[0, {30}]; k = 1; While[k < 2100000001, a = f@ k; If[ t[[a]] == 0, t[[a]] = k; Print[a, " = ", k]]; k++] (* slightly modified by Robert G. Wilson v, Sep 01 2014 *)

a(20)-a(21) from Robert G. Wilson v, Sep 01 2014

A331921 Number of iterations of the map G -> Aut(G) on the cyclic group of order n needed to reach stability; or -1 if the iteration never stabilizes.

Original entry on oeis.org

0, 1, 2, 2, 3, 2, 3, 2, 3, 3, 4, 2, 3, 3, 2, 2, 3, 3, 4, 2, 2, 4, 5, 3, 3, 3, 4, 2, 3, 2, 3

Offset: 1

Sébastien Palcoux, Feb 01 2020

a(n) is the smallest m >= 0 such that Aut^{m+r}(Cn) is isomorphic to Aut^m(Cn) for some r > 0.
This sequence shares the first 7 terms with A003434 but not beyond, because Aut(Cn) has order phi(n) (see A000010) but need not be cyclic. It also shares the first 14 terms with A185816 (not beyond).
For n<32, G=Aut^{m+1}(Cn) is isomorphic to Aut^m(Cn) iff G is in {C1,S3,D8,D12,PGL(2,7)}. This is established by the GAP computation below.
Question: What is a(32)? (we just know that a(32)>=8)
Conjecture: a(n) != -1 for all n.
Question: Is there n such that the sequence Aut^m(Cn) reaches a loop of length>1?
From Jianing Song, Aug 13 2023 and edited on 24 Feb 2025: (Start)
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;
 - p and 2*p for p = 13, 17, 23, 29, 31, 37, 47, 59, 67, 73, 89, 109, 173, 179, 197, 229, 347, 359, 457, 719, 1439, or 2879;
 - divisors of 16, 20, 30, 50, 168, 172, 196, 258, 264, 294, 456, 648, or 686.
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;
 - 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;
 - C_13 or C_26 -> C_8 or C_12 -> C_2 X C_2 -> S_3;
 - 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;
 - C_24 -> C_2 X C_2 X C_2 -> PSL(2,7) -> PGL(2,7);
 - 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;
 - 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;
 - 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;
 - 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);
 - 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;
 - C_168 -> C_2 X C_2 X C_2 X C_6 -> C_2 X A_8 -> S_8;
 - 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);
 - C_264 -> C_2 X C_2 X C_2 X C_10 -> C_4 X A_8 -> C_2 X S_8;
 - 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);
 - 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);
 - 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;
 - 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);
 - C_2879 or C_5758 -> C_1439 or C_2878 -> C_719 or C_1438 -> C_359 or C_718 -> C_179 or C_358 -> 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).
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)?
 - 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)? (End)

			Aut(C1)=C1 so a(1)=0.
Aut(C2)=C1 so a(2)=1.
Aut(C8)=C2xC2, Aut(C2xC2)=S3, Aut(S3)=S3, so a(8)=2 (whereas A003434(8)=3).
Aut(C15)=C2xC4, Aut(C2xC4)=D8, Aut(D8)=D8, so a(15)=2 (whereas A185816(15)=3).

A. P. Goucher, Iterated automorphism groups of finite groups, MathOverflow (2018).
G. Muller, Does Aut(Aut(...Aut(G)...)) stabilize?, MathOverflow (2009).
S. Palcoux, On the iterated automorphism groups of the cyclic groups, MathOverflow (2020).
Wikipedia, Multiplicative group of integers modulo n.

Cf. A000010, A003434, A185816, A117729, A179401.

gap> LoadPackage("sonata");
gap> L:=[];; SG:=[];; for n in [1..31] do a:=0; C:=CyclicGroup(n); A:=AutomorphismGroup(C); while Order(C)<>Order(A) or not IsIsomorphicGroup(A,C) do a:=a+1; C:=A; A:=AutomorphismGroup(A); od; Add(L,a); Add(SG,IdGroup(A)); od;
gap> L;
[ 0, 1, 2, 2, 3, 2, 3, 2, 3, 3, 4, 2, 3, 3, 2, 2, 3, 3, 4, 2, 2, 4, 5, 3, 3, 3, 4, 2, 3, 2, 3 ]
gap> SG;
[ [ 1, 1 ], [ 1, 1 ], [ 1, 1 ], [ 1, 1 ], [ 1, 1 ], [ 1, 1 ], [ 1, 1 ], [ 6, 1 ], [ 1, 1 ], [ 1, 1 ], [ 1, 1 ], [ 6, 1 ], [ 6, 1 ], [ 1, 1 ], [ 8, 3 ], [ 8, 3 ], [ 8, 3 ], [ 1, 1 ], [ 1, 1 ], [ 8, 3 ], [ 12, 4 ], [ 1, 1 ], [ 1, 1 ], [ 336, 208 ], [ 8, 3 ], [ 6, 1 ], [ 1, 1 ], [ 12, 4 ], [ 12, 4 ], [ 8, 3 ], [ 8, 3 ] ]
gap> Set(SG);
[ [ 1, 1 ], [ 6, 1 ], [ 8, 3 ], [ 12, 4 ], [ 336, 208 ] ]
# It is the list of IdGroup for C1, S3, D8, D12 and PGL(2,7).
# The above program works well for n<32. Beyond, it will work as long as there is no loop of length>1 and a(n) finite, which (for small n) is very likely (the opposite would be a breakthrough), otherwise it will just not end. Moreover, if Order(A) is too big then IdGroup(A) will not work, because the SmallGroup library of GAP is finite.

Escape clause added by Jianing Song, Aug 13 2023

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.

A246700 Table read by rows: trajectories under iteration of Carmichael's lambda function (cf. A002322).

Original entry on oeis.org

1, 2, 1, 3, 2, 1, 4, 2, 1, 5, 4, 2, 1, 6, 2, 1, 7, 6, 2, 1, 8, 2, 1, 9, 6, 2, 1, 10, 4, 2, 1, 11, 10, 4, 2, 1, 12, 2, 1, 13, 12, 2, 1, 14, 6, 2, 1, 15, 4, 2, 1, 16, 4, 2, 1, 17, 16, 4, 2, 1, 18, 6, 2, 1, 19, 18, 6, 2, 1, 20, 4, 2, 1, 21, 6, 2, 1, 22, 10, 4, 2, 1

Offset: 1

Reinhard Zumkeller, Sep 02 2014

Length of row n = A185816(n) + 1.

			.   |  1 | 1             | 13 | 13-12-2-1        | 25 | 25-20-4-2-1
.   |  2 | 2-1           | 14 | 14-6-2-1         | 26 | 26-12-2-1
.   |  3 | 3-2-1         | 15 | 15-4-2-1         | 27 | 27-18-6-2-1
.   |  4 | 4-2-1         | 16 | 16-4-2-1         | 28 | 28-6-2-1
.   |  5 | 5-4-2-1       | 17 | 17-16-4-2-1      | 29 | 29-28-6-2-1
.   |  6 | 6-2-1         | 18 | 18-6-2-1         | 30 | 30-4-2-1
.   |  7 | 7-6-2-1       | 19 | 19-18-6-2-1      | 31 | 31-30-4-2-1
.   |  8 | 8-2-1         | 20 | 20-4-2-1         | 32 | 32-8-2-1
.   |  9 | 9-6-2-1       | 21 | 21-6-2-1         | 33 | 33-10-4-2-1
.   | 10 | 10-4-2-1      | 22 | 22-10-4-2-1      | 34 | 34-16-4-2-1
.   | 11 | 11-10-4-2-1   | 23 | 23-22-10-4-2-1   | 35 | 35-12-2-1
.   | 12 | 12-2-1        | 24 | 24-2-1           | 36 | 36-6-2-1  .

Reinhard Zumkeller, Rows n = 1..1001 of triangle, flattened

Cf. A002322, A185816, A375478.

import Data.List (genericIndex)
a246700 n k = genericIndex a246700_tabf (n - 1) !! (k-1)
a246700_row n = genericIndex a246700_tabf (n - 1)
a246700_tabf = [1] : f 2  where
   f x = (x : a246700_row (a002322 x)) : f (x + 1)

Array[Most[FixedPointList[CarmichaelLambda, #]] &, 25] (* Paolo Xausa, Aug 17 2024 *)

T(n,1) = n and T(n,k+1) = A002322(T(n,k)), k = 1..A185816(n).

cp's OEIS Frontend

A027763 Smallest k such that 2^^n is not congruent to 2^^(n-1) mod k, where 2^^n denotes the power tower 2^2^...^2 (in which 2 appears n times).

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A173927 Smallest integer k such that the number of iterations of Carmichael lambda function (A002322) needed to reach 1 starting at k (k is counted) is n.

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A331921 Number of iterations of the map G -> Aut(G) on the cyclic group of order n needed to reach stability; or -1 if the iteration never stabilizes.

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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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A246700 Table read by rows: trajectories under iteration of Carmichael's lambda function (cf. A002322).

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