A332472 The real part of the sum of unitary divisors function (usigma) generalized for Gaussian integers.

1, 1, 4, -3, 4, 4, 8, 1, 10, -12, 12, -12, 6, 8, 16, 17, 6, 10, 20, -12, 32, 12, 24, 4, -24, -30, 28, -24, 8, -48, 32, 1, 48, -38, 32, -30, 8, 20, 24, 68, 10, 32, 44, -36, 40, 24, 48, 68, 50, -40, 24, -18, 10, 28, 48, 8, 80, -64, 60, -48, 12, 32, 80, -63, -120

Offset: 1

Author

Amiram Eldar, Feb 13 2020

Keywords

• sign

Comments

If n = u * Product_{i} p_i^e_i, where u is a unit (1, i, -1 or -i), and p_i is a Gaussian prime with Re(p_i) > 0, then usigma(n) = Product_{i} (p_i^e_i + 1).

a(n) = A103228(n) for odd squarefree numbers (A056911), i.e., numbers n such that A318608(n) != 0.

Examples

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

Links

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

Crossrefs

Cf. A034448, A103228, A332473 (the imaginary 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_] := Re[usigma[n]]; Array[a, 100]