%I #50 Dec 20 2024 11:34:16
%S 5,7,11,13,17,21,23,31,37,40,41,47,53,56,61,67,73,83,88,97,98,101,103,
%T 107,131,136,151,152,156,157,167,173,191,193,223,227,233,237,248,251,
%U 257,263,271,277,296,307,311,328,331,347,353,367,373,376,383,433,443
%N Numbers m that satisfy the equation phi(m + 6) = phi(m) + 6 where phi(m) = A000010(m) is Euler's totient function.
%C The majority of solutions can be predicted by known properties of the equality. There are several solutions that do not fit these parameters.
%C An odd natural number m is a solution if m and m + 6 are both prime (sexy primes) (A023201).
%C Among the solutions for even natural numbers are all m = 8*p with odd primes p such that 4*p+3 is a prime number. Proof: From A000010 we can learn that the formula phi(p*2) = floor(((2 + p - 1) mod p)/(p - 1)) + p - 1 is known. If we define p = 4*q+3 and m = 8*q and insert, we will obtain phi(8*q+6) = 4*q+2. Also it is known that phi(8*q) = 4*q-4 if q is any odd prime. - _Thomas Scheuerle_, Dec 20 2024
%H Peter Kagey, <a href="/A262084/b262084.txt">Table of n, a(n) for n = 1..10000</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Euler%27s_totient_function">Euler's totient function</a>
%e 5 is a term since phi(5+6) = 10 = 6 + 4 = phi(5) + 6.
%t Select[Range@500, EulerPhi@(# + 6)== EulerPhi[#] + 6 &] (* _Vincenzo Librandi_, Sep 11 2015 *)
%o (Sage) [n for n in [1..1000] if euler_phi(n+6)==euler_phi(n)+6] # _Tom Edgar_, Sep 10 2015
%o (Magma) [n: n in [1..500] | EulerPhi(n+6) eq EulerPhi(n)+6]; // _Vincenzo Librandi_, Sep 11 2015
%o (PARI) is(n)=eulerphi(n + 6) == eulerphi(n) + 6 \\ _Anders Hellström_, Sep 11 2015
%Y Cf. A000010, A023201.
%Y Cf. A001838 (k=2), A056772 (k=4), A262085 (k=8), A262086 (k=10).
%K nonn,easy
%O 1,1
%A _Kevin J. Gomez_, Sep 10 2015