A103230 Norm of the sum of divisors function sigma(n) generalized for Gaussian integers.

1, 13, 16, 41, 80, 208, 64, 113, 169, 1040, 144, 656, 360, 832, 1280, 481, 520, 2197, 400, 3280, 1024, 1872, 576, 1808, 2257, 4680, 1600, 2624, 1360, 16640, 1024, 2113, 2304, 6760, 5120, 6929, 2000, 5200, 5760, 9040, 2600, 13312, 1936, 5904, 13520

Offset: 1

Author

T. D. Noe, Jan 26 2005

Keywords

• nonn
• mult

Comments

See A102506 for a complete description.

See A103228 and A103229 for the real and imaginary parts.

Multiplicative because the sigma function on Gaussian integers as defined in A102506 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 Andrew Howroyd)

Crossrefs

Cf. A102506, A102574, A103224, A103228, A103229.

Programs

Mathematica

Abs[Table[DivisorSigma[1, n, GaussianIntegers -> True], {n, 100}]]^2

PARI

\\ See A102506 for formula.
CSigma(z)={my(f=factor(z,I)); prod(i=1, #f~, my([p,e]=f[i,]); if(norm(p)==1, 1, (p^(e+1)-1)/(p-1)))}
a(n)=norm(CSigma(n)); \\ Andrew Howroyd, Aug 03 2018

Formula

a(n) = A103228(n)^2 + A103229(n)^2. - Andrew Howroyd, Aug 03 2018