A281016 Numbers n such that n, phi(n) and cototient(n) are all perfect powers.

8, 16, 32, 64, 125, 128, 256, 512, 1024, 2048, 3125, 4096, 4913, 8192, 16384, 32768, 50653, 65536, 78125, 131072, 262144, 524288, 1030301, 1048576, 1419857, 1953125, 2097152, 4194304, 7645373, 8388608, 16777216, 16974593, 33554432, 35831808, 48828125, 64481201, 67108864, 69343957

Offset

1,1

Comments

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.

Links

Table of n, a(n) for n=1..38.

Example

125 = 5^3 is a term because phi(5^3) = 10^2 and cototient(5^3) = 5^2.

Mathematica

Select[Range[10^6], Times @@ Boole@ Map[Or[# == 1, GCD @@ FactorInteger[#][[All, 2]] > 1] &, {#, EulerPhi@ #, # - EulerPhi@ #}] > 0 &] (* Michael De Vlieger, Jan 14 2017 *)

Prog

(PARI) is(n) = ispower(eulerphi(n)) && ispower(n-eulerphi(n)) && ispower(n);

Crossrefs

Cf. A000010, A001597, A051953, A054754, A166955.

Sequence in context: A192135 A345053 A256817 * A048169 A089882 A335850

Adjacent sequences: A281013 A281014 A281015 * A281017 A281018 A281019

Keyword

nonn

Author

Altug Alkan, Jan 13 2017

Status

approved