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%I #23 Apr 19 2021 17:12:43
%S 1,2,3,4,5,6,7,9,10,11,14,18,19,22,23,27,38,46,47,54,81,94,162,163,
%T 243,326,486,487,729,974,1458,1459,2187,2918,4374,6561,13122,19683,
%U 39366,39367,59049,78734,118098,177147,354294,531441,1062882,1594323,3188646,4782969
%N 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.
%C If the map "G -> Automorphism group of G" eventually reaches the trivial group, then the initial group IS a cyclic group.
%C From _Jianing Song_, Oct 12 2019: (Start)
%C 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.
%C Number of iterations to reach the trivial group:
%C k = 1: 0;
%C k = 2: 1;
%C k = 4: 2;
%C k = 5, 10: 3;
%C k = 11, 22: 4;
%C k = 23, 46: 5;
%C k = 47, 94: 6;
%C k = 3^i, 2*3^i, i > 0: i+1;
%C k = 2*3^i+1, 2*(2*3^i+1), i > 0, 2*3^i+1 is prime: i+2. (End)
%C From _Peter Schorn_, Apr 06 2021: (Start)
%C 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.
%C 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.
%C (End)
%F Consists of the following numbers:
%F 3^i and 2*3^i for all i >= 0;
%F if 2*3^i+1 is a prime, then also 2*3^i+1 and 2(2*3^i+1);
%F the exceptional entries 4, 5, 10, 11, 22, 23, 46, 47 and 94.
%p 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(%);
%o (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}
%o { for(n=1, 10^9, if(ok(n), print1(n, ", "))) } \\ _Andrew Howroyd_, Oct 12 2019
%Y Cf. A033948, A000010, A258615, A331921, A003434, A185816.
%K nonn
%O 1,2
%A _N. J. A. Sloane_, based on a communication from _J. H. Conway_, Apr 14 2006
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