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