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Numbers k such that phi(k) is a perfect 5th power.
8

%I #25 Sep 27 2022 04:03:43

%S 1,2,51,64,68,80,96,102,120,1285,2048,2056,2176,2560,2570,2720,3072,

%T 3084,3264,3840,4080,7957,8227,8279,9079,9139,9709,9919,10355,10595,

%U 11667,11673,11691,12099,12393,12483,12753,12987,13797,14715,14763

%N Numbers k such that phi(k) is a perfect 5th power.

%C As phi(2^(5*n+1)) = (2^n)^5, A013822 is a subsequence. - _Bernard Schott_, Sep 26 2022

%C Numbers of the form u = 2^(5*k)*3^(5*m + 1), k>=1, m>=0, are terms because phi(u) = 2^(5*k)*3^(5*m) = (2^k*3^m)^5. - _Marius A. Burtea_, Sep 26 2022

%H Amiram Eldar, <a href="/A078165/b078165.txt">Table of n, a(n) for n = 1..10000</a>

%e phi of the sequence includes 1, 32, 1024, 7776, ...; powers arise several times; a(3) = A053576(5) = 51.

%t k=5; Do[s=EulerPhi[n]^(1/k); If[IntegerQ[s], Print[n]], {n, 1, 5000}]

%t Select[Range[15000],IntegerQ[Surd[EulerPhi[#],5]]&] (* _Harvey P. Dale_, Jul 26 2019 *)

%o (PARI) is(n)=ispower(eulerphi(n),5) \\ _Charles R Greathouse IV_, Apr 24 2020

%Y A013822 is a subsequence.

%Y Cf. A039770 (square), A039771 (cube), A078164 (4th), A078165 (5th, this sequence), A078166 (6th), A078167 (7th), A078168 (8th), A078169 (9th), A078170 (10th power), A001317, A053576, A045544, A000010.

%K nonn

%O 1,2

%A _Labos Elemer_, Nov 27 2002