A171271 Numbers n such that phi(n)=2*phi(n-1).

3, 5, 17, 155, 257, 287, 365, 805, 1067, 2147, 3383, 4551, 6107, 7701, 8177, 9269, 11285, 12557, 12971, 16403, 19229, 19277, 20273, 25133, 26405, 27347, 29155, 29575, 35645, 36419, 38369, 39647, 40495, 47215, 52235, 54653, 65537, 84863

Offset

1,1

Comments

Theorem: A prime p is in the sequence iff p is a Fermat prime.

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.

Links

Giovanni Resta, Table of n, a(n) for n = 1..10000 (first 5416 terms from Hiroaki Yamanouchi)

Formula

a(n) = A050472(n) + 1. - Ray Chandler, May 01 2015

Mathematica

Select[Range[85000], EulerPhi[ # ]==2EulerPhi[ #-1]&]
Flatten[Position[Partition[EulerPhi[Range[90000]], 2, 1], _?(2#[[1]] == #[[2]]&), 1, Heads->False]]+1 (* Harvey P. Dale, Sep 09 2017 *)

Prog

(Magma) [n: n in [2..2*10^5] | EulerPhi(n) eq 2*EulerPhi(n-1)]; // Vincenzo Librandi, May 17 2015

Crossrefs

Cf. A019434, A050472, A171262.

Sequence in context: A348430 A107312 A083213 * A056826 A278138 A273870

Adjacent sequences: A171268 A171269 A171270 * A171272 A171273 A171274

Keyword

easy,nonn

Author

Farideh Firoozbakht, Feb 23 2010

Status

approved