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

Author

Farideh Firoozbakht, Feb 23 2010

Keywords

• easy
• nonn

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)

Crossrefs

Cf. A019434, A050472, A171262.

Programs

Magma

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

Formula

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