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%I #14 Sep 08 2022 08:45:19
%S 1,4,8,16,32,64,110,128,220,256,440,506,512,550,880,1012,1024,1100,
%T 1760,1830,2024,2048,2162,2200,2750,3422,3520,3660,4048,4096,4114,
%U 4324,4400,4746,5490,5500,5566,6806,6844,7040,7320,7782,8096,8192,8228,8648,8800,9150,9492
%N Numbers n such that phi(n) = phi(n + phi(n)).
%C If n is an even term of this sequence then 2n is also in the sequence. This is because phi(2n) = 2*phi(n) = 2*phi(n+phi(n)) = phi(2n+ 2*phi(n)) = phi(2n+phi(2n)). If n is an even term of this sequence then for each natural number m, 2^m*n is in the sequence. For example, since 4 is in the sequence 2^n for each n, n>1 is in the sequence. If p is a Sophie Germain prime greater than 3 then n = 2*p*(2p+1) is in the sequence because phi(n+phi(n)) = phi(2*p*(2p+1)+2*p*(p-1)) = phi(6p^2) = 2*p*(p-1) = phi(n). Conjecture: Except for the first term all terms are even.
%C If n is in the sequence and the natural number m divides gcd(phi(n),n) then for all nonnegative integers k, m^k*n are in the sequence. For example 110 is in the sequence and 10 divides gcd(phi(110),110), so 11*10^k for all natural numbers k are in the sequence. - _Farideh Firoozbakht_, Dec 12 2005
%H Vincenzo Librandi, <a href="/A108569/b108569.txt">Table of n, a(n) for n = 1..384</a>
%p with(numtheory): A108569:=n->`if`(phi(n) = phi(n+phi(n)), n, NULL): seq(A108569(n), n=1..10^4); # _Wesley Ivan Hurt_, Nov 12 2014
%t Select[Range[11000], EulerPhi[ # ]==EulerPhi[ # + EulerPhi[ # ]]&]
%o (Magma) [n: n in [1..10000] | EulerPhi(n) eq EulerPhi(n + EulerPhi(n))]; // _Vincenzo Librandi_, Nov 13 2014
%o (PARI) select(n->eulerphi(n) == eulerphi(n + eulerphi(n)), vector(10000, i, i)) \\ _Michel Marcus_, Nov 13 2014
%Y Cf. A005384, A051487.
%K nonn
%O 1,2
%A _Farideh Firoozbakht_, Jul 05 2005
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