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.

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

Author

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

Keywords

• nonn

Comments

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)

Crossrefs

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

Programs

Maple

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(%);

PARI

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

Formula

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.