A103225 Number of Gaussian integers z with abs(z) < n and gcd(n,z)=1.

1, 4, 24, 24, 44, 48, 144, 96, 224, 96, 372, 192, 444, 304, 404, 392, 792, 448, 1124, 408, 1200, 752, 1648, 808, 1240, 896, 2036, 1200, 2440, 800, 2996, 1600, 3008, 1592, 2404, 1808, 4056, 2256, 3616, 1600, 4992, 2400, 5784, 3008, 3604, 3304, 6916, 3224, 7376

Offset: 1

Author

T. D. Noe, Jan 26 2005

Keywords

• nice
• nonn

Comments

This sequence is much like the usual totient function. That is, it gives the number of Gaussian integers that are relatively prime to n and whose modulus is less than n. When n is a Gaussian prime, A002145, then a(n) = A051132(n)-1.

Four of the dominant lines of the plot appear to align to k(i)*Pi*n^2, with k(i) = 1, 8/9, 1/2, and 4/9. Conjecture: a(n) < Pi*n^2. - Bill McEachen, Aug 14 2025

Examples

			a(2)=4 because 1, -1, i and -i are relatively prime to 2 and have modulus less than 2.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..1000 (terms 1..200 from T. D. Noe)

Crossrefs

Cf. A002145, A051132, A103222, A103223, A103224.

Programs

Mathematica

Table[cnt=0; Do[z=a+ b*I; If[Abs[z]