A333909 Numbers k such that phi(k) is the sum of 2 squares, where phi is the Euler totient function (A000010).
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, 40, 41, 44, 48, 50, 51, 53, 54, 55, 57, 59, 60, 63, 64, 66, 68, 73, 74, 75, 76, 80, 82, 83, 85, 88, 91, 95, 96, 100, 101, 102, 106, 107, 108, 110, 111, 114, 117, 118, 120
Offset: 1
Keywords
Examples
1 is a term since phi(1) = 1 = 0^2 + 1^2.
Links
- Amiram Eldar, Table of n, a(n) for n = 1..10000
- William D. Banks, Florian Luca, Filip Saidak, and Igor E. Shparlinski, Values of arithmetical functions equal to a sum of two squares, Quarterly Journal of Mathematics, Vol. 56, No. 2 (2005), pp. 123-139, alternative link.
Programs
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Mathematica
Select[Range[120], SquaresR[2, EulerPhi[#]] > 0 &]
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Python
from itertools import count, islice from sympy import factorint, totient def A333909_gen(): # generator of terms return filter(lambda n:all(p & 3 != 3 or e & 1 == 0 for p, e in factorint(totient(n)).items()),count(1)) A333909_list = list(islice(A333909_gen(),30)) # Chai Wah Wu, Jun 27 2022
Formula
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).
Comments