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A332472 The real part of the sum of unitary divisors function (usigma) generalized for Gaussian integers. 6

%I

%S 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,

%T -30,28,-24,8,-48,32,1,48,-38,32,-30,8,20,24,68,10,32,44,-36,40,24,48,

%U 68,50,-40,24,-18,10,28,48,8,80,-64,60,-48,12,32,80,-63,-120

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

%C 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).

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

%H Amiram Eldar, <a href="/A332472/b332472.txt">Table of n, a(n) for n = 1..10000</a>

%e 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.

%t 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]

%Y Cf. A034448, A103228, A332473 (the imaginary part), A332474 (the norm).

%K sign

%O 1,3

%A _Amiram Eldar_, Feb 13 2020

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Last modified August 17 23:13 EDT 2022. Contains 356204 sequences. (Running on oeis4.)