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Numbers k such that phi(k) and phi(k+1) are perfect powers (A001597).
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%I #16 Sep 08 2022 08:46:25

%S 1,15,16,63,101,125,255,256,272,504,512,513,629,679,1358,1359,1728,

%T 1970,2047,2222,2509,2834,3458,3705,4094,4095,4400,4577,4616,4913,

%U 5403,6817,7295,7956,8729,11667,11672,16132,16384,16523,17507,23085,24198,24564,24624,25220,25601,27216,27384,28564,29444

%N Numbers k such that phi(k) and phi(k+1) are perfect powers (A001597).

%e phi(101) = 10^2, and phi(102) = 2^5.

%e phi(3458) = 6^4, and phi(3459) = 48^2.

%t perfectPowerQ[1] = True; perfectPowerQ[n_] := GCD @@ FactorInteger[n][[;; , 2]] > 1; Select[Range[30000], And @@ perfectPowerQ /@ EulerPhi[# + {0, 1}] &] (* _Amiram Eldar_, Feb 04 2020 *)

%o (PARI) v=[1]; for(i = 2, 30000, if(ispower(eulerphi(i)), if(ispower(eulerphi(i+1)), v = concat(v, i)))); v

%o (Magma) [1] cat [k:k in [3..30000]|IsPower(EulerPhi(k)) and IsPower(EulerPhi(k+1))]; // _Marius A. Burtea_, Feb 05 2020

%Y Cf. A000010, A001597, A166955, A281016.

%K nonn

%O 1,2

%A _Antonio Roldán_, Feb 04 2020