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%I #21 Sep 08 2022 08:45:49
%S 3,5,17,155,257,287,365,805,1067,2147,3383,4551,6107,7701,8177,9269,
%T 11285,12557,12971,16403,19229,19277,20273,25133,26405,27347,29155,
%U 29575,35645,36419,38369,39647,40495,47215,52235,54653,65537,84863
%N Numbers n such that phi(n)=2*phi(n-1).
%C Theorem: A prime p is in the sequence iff p is a Fermat prime.
%C Proof: If p=2^2^n+1 is prime (Fermat prime) then phi(p)=2^2^n=2* phi(2^2^n)=2*phi(p-1), so p is in the sequence. Now if p is a prime term of the sequence then phi(p)=2*phi(p-1) so p-1=2*phi(p-1) and we deduce that p-1=2^m hence p is a Fermat prime.
%H Giovanni Resta, <a href="/A171271/b171271.txt">Table of n, a(n) for n = 1..10000</a> (first 5416 terms from Hiroaki Yamanouchi)
%F a(n) = A050472(n) + 1. - _Ray Chandler_, May 01 2015
%t Select[Range[85000],EulerPhi[ # ]==2EulerPhi[ #-1]&]
%t Flatten[Position[Partition[EulerPhi[Range[90000]],2,1],_?(2#[[1]] == #[[2]]&),1,Heads->False]]+1 (* _Harvey P. Dale_, Sep 09 2017 *)
%o (Magma) [n: n in [2..2*10^5] | EulerPhi(n) eq 2*EulerPhi(n-1)]; // _Vincenzo Librandi_, May 17 2015
%Y Cf. A019434, A050472, A171262.
%K easy,nonn
%O 1,1
%A _Farideh Firoozbakht_, Feb 23 2010