A103224 Norm of the totient function phi(n) for Gaussian integers. See A103222 and A103223 for the real and imaginary parts.

1, 2, 4, 8, 8, 8, 36, 32, 36, 16, 100, 32, 80, 72, 32, 128, 160, 72, 324, 64, 144, 200, 484, 128, 200, 160, 324, 288, 520, 64, 900, 512, 400, 320, 288, 288, 936, 648, 320, 256, 1088, 288, 1764, 800, 288, 968, 2116, 512, 1764, 400, 640, 640, 2000, 648, 800, 1152

Offset

1,2

Comments

See A103222 for definitions.

Multiplicative because the totient function on Gaussian integers is multiplicative and the norm is completely multiplicative. - Andrew Howroyd, Aug 03 2018

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)

Formula

a(n) = A103222(n)^2 + A103223(n)^2. - Andrew Howroyd, Aug 03 2018

Mathematica

phi[z_] := Module[{f, k, prod}, If[Abs[z]==1, z, f=FactorInteger[z, GaussianIntegers->True]; If[Abs[f[[1, 1]]]==1, k=2; prod=f[[1, 1]], k=1; prod=1]; Do[prod=prod*(f[[i, 1]]-1)f[[i, 1]]^(f[[i, 2]]-1), {i, k, Length[f]}]; prod]]; Abs[Table[phi[n], {n, 100}]]^2

Prog

(PARI) \\ See A103222
CEulerPhi(z)={my(f=factor(z, I)); prod(i=1, #f~, my([p, e]=f[i, ]); if(norm(p)==1, p^e, (p-1)*p^(e-1)))}
a(n)=norm(CEulerPhi(n)); \\ Andrew Howroyd, Aug 03 2018

Crossrefs

Cf. A103222, A103223, A103230.

Sequence in context: A366569 A187221 A129280 * A198346 A078750 A360156

Adjacent sequences: A103221 A103222 A103223 * A103225 A103226 A103227

Keyword

nonn,mult

Author

T. D. Noe, Jan 26 2005

Status

approved