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A281016
Numbers n such that n, phi(n) and cototient(n) are all perfect powers.
1
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.
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
KEYWORD
nonn
AUTHOR
Altug Alkan, Jan 13 2017
STATUS
approved