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A078164
Numbers k such that phi(k) is a perfect biquadrate.
19
1, 2, 17, 32, 34, 40, 48, 60, 257, 512, 514, 544, 640, 680, 768, 816, 960, 1020, 1297, 1387, 1417, 1729, 1971, 2109, 2223, 2289, 2331, 2445, 2457, 2565, 2594, 2608, 2774, 2812, 2834, 2835, 3052, 3260, 3458, 3888, 3912, 3924, 3942, 3996, 4104, 4212, 4218
OFFSET
1,2
COMMENTS
Corresponding values of phi include 1, 16, 256, 1296, 4096, ... and these arise several times each.
a(3) = A053576(4).
A013776 is a subsequence since phi(2^(4*n+1)) = (2^n)^4. - Bernard Schott, Sep 22 2022
Subsequence of primes is A037896 since in this case: phi(k^4+1) = k^4. - Bernard Schott, Mar 05 2023
LINKS
MATHEMATICA
k=4; Do[s=EulerPhi[n]^(1/k); If[IntegerQ[s], Print[n]], {n, 1, 5000}]
Select[Range[5000], IntegerQ[Surd[EulerPhi[#], 4]]&] (* Harvey P. Dale, Apr 30 2015 *)
PROG
(PARI) is(n)=ispower(eulerphi(n), 4) \\ Charles R Greathouse IV, Apr 24 2020
(Python)
from itertools import count, islice
from sympy import totient, integer_nthroot
def A078164_gen(startvalue=1): # generator of terms >= startvalue
return filter(lambda n:integer_nthroot(totient(n), 4)[1], count(max(1, startvalue)))
A078164_list = list(islice(A078164_gen(), 20)) # Chai Wah Wu, Feb 28 2023
CROSSREFS
Subsequence of A039770. A037896 is a subsequence.
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).
Sequence in context: A267540 A141068 A162622 * A060387 A003336 A344187
KEYWORD
nonn
AUTHOR
Labos Elemer, Nov 27 2002
STATUS
approved