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A333909 Numbers k such that phi(k) is the sum of 2 squares, where phi is the Euler totient function (A000010).

%I A333909 #18 Jun 27 2022 23:49:14
%S A333909 1,2,3,4,5,6,8,10,11,12,15,16,17,19,20,22,24,25,27,30,32,33,34,37,38,
%T A333909 40,41,44,48,50,51,53,54,55,57,59,60,63,64,66,68,73,74,75,76,80,82,83,
%U A333909 85,88,91,95,96,100,101,102,106,107,108,110,111,114,117,118,120
%N A333909 Numbers k such that phi(k) is the sum of 2 squares, where phi is the Euler totient function (A000010).
%H A333909 Amiram Eldar, <a href="/A333909/b333909.txt">Table of n, a(n) for n = 1..10000</a>
%H A333909 William D. Banks, Florian Luca, Filip Saidak, and Igor E. Shparlinski, <a href="https://doi.org/10.1093/qmath/hah039">Values of arithmetical functions equal to a sum of two squares</a>, Quarterly Journal of Mathematics, Vol. 56, No. 2 (2005), pp. 123-139, <a href="https://faculty.missouri.edu/~bankswd/papers/2005_arith_functs_sum_two_squares.pdf">alternative link</a>.
%F A333909 c1 * x/log(x)^(3/2) < N(x) < c2 * x/log(x)^(3/2), where N(x) is the number of terms <= x, and c1 and c2 are two positive constants (Banks et al., 2005).
%e A333909 1 is a term since phi(1) = 1 = 0^2 + 1^2.
%t A333909 Select[Range[120], SquaresR[2, EulerPhi[#]] > 0 &]
%o A333909 (Python)
%o A333909 from itertools import count, islice
%o A333909 from sympy import factorint, totient
%o A333909 def A333909_gen(): # generator of terms
%o A333909     return filter(lambda n:all(p & 3 != 3 or e & 1 == 0 for p, e in factorint(totient(n)).items()),count(1))
%o A333909 A333909_list = list(islice(A333909_gen(),30)) # _Chai Wah Wu_, Jun 27 2022
%Y A333909 Cf. A000010, A001481, A039770, A079545, A333910, A333911, A333912.
%K A333909 nonn
%O A333909 1,2
%A A333909 _Amiram Eldar_, Apr 09 2020