A108569 Numbers n such that phi(n) = phi(n + phi(n)).

1, 4, 8, 16, 32, 64, 110, 128, 220, 256, 440, 506, 512, 550, 880, 1012, 1024, 1100, 1760, 1830, 2024, 2048, 2162, 2200, 2750, 3422, 3520, 3660, 4048, 4096, 4114, 4324, 4400, 4746, 5490, 5500, 5566, 6806, 6844, 7040, 7320, 7782, 8096, 8192, 8228, 8648, 8800, 9150, 9492

Offset: 1

Author

Farideh Firoozbakht, Jul 05 2005

Keywords

• nonn

Comments

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.

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

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..384

Crossrefs

Cf. A005384, A051487.

Programs

Magma

[n: n in [1..10000] | EulerPhi(n) eq EulerPhi(n + EulerPhi(n))]; // Vincenzo Librandi, Nov 13 2014

Maple

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

Mathematica

Select[Range[11000], EulerPhi[ # ]==EulerPhi[ # + EulerPhi[ # ]]&]

PARI

select(n->eulerphi(n) == eulerphi(n + eulerphi(n)), vector(10000, i, i)) \\ Michel Marcus, Nov 13 2014