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%I #13 Jan 20 2017 13:28:42
%S 8,16,32,64,125,128,256,512,1024,2048,3125,4096,4913,8192,16384,32768,
%T 50653,65536,78125,131072,262144,524288,1030301,1048576,1419857,
%U 1953125,2097152,4194304,7645373,8388608,16777216,16974593,33554432,35831808,48828125,64481201,67108864,69343957
%N Numbers n such that n, phi(n) and cototient(n) are all perfect powers.
%C This sequence does not contain only prime powers. Least term that has a prime factor which is not of the form m^2 + 1 is 35831808 = 2^14 * 3^7. The next one is 102503232 = 2^6 * 3^6 * 13^3. There are infinitely many such numbers.
%e 125 = 5^3 is a term because phi(5^3) = 10^2 and cototient(5^3) = 5^2.
%t Select[Range[10^6], Times @@ Boole@ Map[Or[# == 1, GCD @@ FactorInteger[#][[All, 2]] > 1] &, {#, EulerPhi@ #, # - EulerPhi@ #}] > 0 &] (* _Michael De Vlieger_, Jan 14 2017 *)
%o (PARI) is(n) = ispower(eulerphi(n)) && ispower(n-eulerphi(n)) && ispower(n);
%Y Cf. A000010, A001597, A051953, A054754, A166955.
%K nonn
%O 1,1
%A _Altug Alkan_, Jan 13 2017
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