A257865 - OEIS

%I #23 Jun 05 2015 08:28:46

%S 1,5,119,629,17907119

%N Smallest k such that phi(k) = n*phi(k+1), where phi(n) = A000010(n) gives the value of Euler's totient function at n.

%C From _Manfred Scheucher_, May 27 2015: (Start)

%C a(6)>=3*10^8 (calculation)

%C a(7)>=3.5*10^13, a(8)>=4.5*10^25, a(9)>=3.0*10^47, and so on... (doubly exponential lower bound, see uploaded pdf)

%C 239719159679 and 239742643139 admit a ratio of 5.998... and 6.008..., resp.

%C There might be a relation to the sequence A098026. (End)

%H Manfred Scheucher, <a href="/A257865/a257865.pdf">A lower bound on A257865(n)</a>

%F a(n) >= exp(exp(c(n-3))) with c=exp(gamma) and gamma being the Euler-Mascheroni_constant (see pdf). - _Manfred Scheucher_, May 27 2015

%e a(3) = 119, because phi(119) == 3*phi(120) = 96 and 119 is the smallest k where this equality holds for n = 3.

%t Table[k = 1; While[EulerPhi[k] != n EulerPhi[k + 1], k++]; k, {n, 4}] (* _Michael De Vlieger_, May 12 2015 *)

%o (PARI) a(n) = my(k=1); while(eulerphi(k)!=n*eulerphi(k+1), k++); k

%Y Cf. A001274, A171262, A256907, A256937, A257550.

%K nonn,more

%O 1,2

%A _Felix Fröhlich_, May 11 2015