A332473 The imaginary part of the sum of unitary divisors function (usigma) generalized for Gaussian integers.

0, 2, 0, 0, 8, 8, 0, -8, 0, 16, 0, 0, 18, 16, 32, 0, 22, 20, 0, -24, 0, 24, 0, -32, 8, 30, 0, 0, 36, 64, 0, 32, 0, 34, 64, 0, 44, 40, 72, -24, 50, 64, 0, 0, 80, 48, 0, 0, 0, -40, 88, -54, 62, 56, 96, -64, 0, 52, 0, -96, 72, 64, 0, 0, 120, 96, 0, -66, 0, 128, 0

Offset: 1

Author

Amiram Eldar, Feb 13 2020

Keywords

• sign

Comments

See A332472 for a description.

Examples

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

Links

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

Crossrefs

Cf. A034448, A103229, A332472 (the real part), A332474 (the norm).

Programs

Mathematica

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