A078165 Numbers k such that phi(k) is a perfect 5th power.

1, 2, 51, 64, 68, 80, 96, 102, 120, 1285, 2048, 2056, 2176, 2560, 2570, 2720, 3072, 3084, 3264, 3840, 4080, 7957, 8227, 8279, 9079, 9139, 9709, 9919, 10355, 10595, 11667, 11673, 11691, 12099, 12393, 12483, 12753, 12987, 13797, 14715, 14763

Offset: 1

Author

Labos Elemer, Nov 27 2002

Keywords

• nonn

Comments

As phi(2^(5*n+1)) = (2^n)^5, A013822 is a subsequence. - Bernard Schott, Sep 26 2022

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

Examples

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

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

A013822 is a subsequence.

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.

Programs

Mathematica

k=5; Do[s=EulerPhi[n]^(1/k); If[IntegerQ[s], Print[n]], {n, 1, 5000}]
Select[Range[15000],IntegerQ[Surd[EulerPhi[#],5]]&] (* Harvey P. Dale, Jul 26 2019 *)

PARI

is(n)=ispower(eulerphi(n),5) \\ Charles R Greathouse IV, Apr 24 2020