A332474 The norm of the sum of unitary divisors function (usigma) generalized for Gaussian integers.

1, 5, 16, 9, 80, 80, 64, 65, 100, 400, 144, 144, 360, 320, 1280, 289, 520, 500, 400, 720, 1024, 720, 576, 1040, 640, 1800, 784, 576, 1360, 6400, 1024, 1025, 2304, 2600, 5120, 900, 2000, 2000, 5760, 5200, 2600, 5120, 1936, 1296, 8000, 2880, 2304, 4624, 2500, 3200

Offset: 1

Author

Amiram Eldar, Feb 13 2020

Keywords

• nonn

Comments

See A332472 for a description.

Examples

			a(4) = 9 since 4 = -(1 + i)^4 in Gaussian integers (i is the imaginary unit), so usigma(4) = (1 + i)^4 + 1 = -3, and a(4) = (-3)^2 + 0^2 = 9.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A034448, A103230, A332472 (the real part), A332473 (the imaginary part).

Programs

Mathematica

f[p_, e_] := If[Abs[p] == 1, 1, (p^e + 1)]; usigma[n_] := Times @@ f @@@ FactorInteger[n, GaussianIntegers -> True]; a[n_] := Abs[usigma[n]]^2; Array[a, 100]

Formula

a(n) = A332472(n)^2 + A332473(n)^2.