This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A171262 #11 Sep 08 2022 08:45:49 %S A171262 5,13,35,37,61,73,157,193,277,313,397,421,455,457,541,613,661,665,673, %T A171262 733,757,877,997,1085,1093,1153,1201,1213,1237,1295,1321,1381,1453, %U A171262 1621,1657,1753,1873,1933,1993,2017,2137,2169,2341,2473,2557,2593,2797,2857 %N A171262 Numbers n such that phi(n) = 2*phi(n+1). %C A171262 Theorem: A prime p is in the sequence iff 1/2*(p+1) is prime. %C A171262 Proof: If both numbers p & 1/2*(p+1) are prime then phi(p)=p-1=2*(p-1)/2 %C A171262 2*(1/2*(p+1)-1)=2*phi(1/2*(p+1)), 1/2*(p+1) is odd so phi(1/2*(p+1))= %C A171262 phi(p+1) hence phi(p)=2*phi(p+1), namely p is in the sequence. %C A171262 Also if p is a prime term of the sequence %C A171262 then phi(p)=2*phi(p+1) so %C A171262 p-1=2*phi(p+1) or phi(p+1)=1/2*(p+1)-1 hence 1/2*(p+1)is prime. %H A171262 Ray Chandler, <a href="/A171262/b171262.txt">Table of n, a(n) for n = 1..10000</a> %F A171262 phi(35)=2*12=2*phi(35+1), so 35 is in the sequence. %t A171262 Select[Range[2900],EulerPhi[ # ]==2EulerPhi[ #+1]&] %o A171262 (Magma) [n: n in [1..3*10^3] | EulerPhi(n) eq 2*EulerPhi(n+1)]; // _Vincenzo Librandi_, Apr 14 2015 %Y A171262 Cf. A005383, A171271. %K A171262 nonn,easy %O A171262 1,1 %A A171262 _Farideh Firoozbakht_, Feb 23 2010