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A078164 Numbers k such that phi(k) is a perfect biquadrate.

%I A078164 #45 Jun 02 2024 08:50:10
%S A078164 1,2,17,32,34,40,48,60,257,512,514,544,640,680,768,816,960,1020,1297,
%T A078164 1387,1417,1729,1971,2109,2223,2289,2331,2445,2457,2565,2594,2608,
%U A078164 2774,2812,2834,2835,3052,3260,3458,3888,3912,3924,3942,3996,4104,4212,4218
%N A078164 Numbers k such that phi(k) is a perfect biquadrate.
%C A078164 Corresponding values of phi include 1, 16, 256, 1296, 4096, ... and these arise several times each.
%C A078164 a(3) = A053576(4).
%C A078164 A013776 is a subsequence since phi(2^(4*n+1)) = (2^n)^4. - _Bernard Schott_, Sep 22 2022
%C A078164 Subsequence of primes is A037896 since in this case: phi(k^4+1) = k^4. - _Bernard Schott_, Mar 05 2023
%H A078164 Amiram Eldar, <a href="/A078164/b078164.txt">Table of n, a(n) for n = 1..10000</a>
%t A078164 k=4; Do[s=EulerPhi[n]^(1/k); If[IntegerQ[s], Print[n]], {n, 1, 5000}]
%t A078164 Select[Range[5000],IntegerQ[Surd[EulerPhi[#],4]]&] (* _Harvey P. Dale_, Apr 30 2015 *)
%o A078164 (PARI) is(n)=ispower(eulerphi(n),4) \\ _Charles R Greathouse IV_, Apr 24 2020
%o A078164 (Python)
%o A078164 from itertools import count, islice
%o A078164 from sympy import totient, integer_nthroot
%o A078164 def A078164_gen(startvalue=1): # generator of terms >= startvalue
%o A078164     return filter(lambda n:integer_nthroot(totient(n),4)[1], count(max(1,startvalue)))
%o A078164 A078164_list = list(islice(A078164_gen(),20)) # _Chai Wah Wu_, Feb 28 2023
%Y A078164 Subsequence of A039770. A037896 is a subsequence.
%Y A078164 Sequences where phi(k) is a perfect power: A039770 (square), A039771 (cube), this sequence (4th), A078165 (5th), A078166 (6th), A078167 (7th), A078168 (8th), A078169 (9th), A078170 (10th).
%Y A078164 Cf. A001317, A053576, A037896, A045544, A000010, A013776.
%K A078164 nonn
%O A078164 1,2
%A A078164 _Labos Elemer_, Nov 27 2002