

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.


2



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
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OFFSET

1,2


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)


LINKS

Table of n, a(n) for n=1..50.


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.


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(%);


PROG

(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


CROSSREFS

Cf. A033948, A000010, A258615, A331921, A003434, A185816.
Sequence in context: A331802 A271108 A179401 * A073726 A194125 A008839
Adjacent sequences: A117726 A117727 A117728 * A117730 A117731 A117732


KEYWORD

nonn


AUTHOR

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


STATUS

approved



