Numbers n such that for the Gaussian integer z = n + n*I, sigma(z)/ z is a Gaussian integer.
5, 10, 120, 4680, 6240, 31824000, 628992000
1
The Gaussian integer z is called a complex multiperfect number.
T. D. Noe, Plot of 7 terms
Robert Spira, The complex sum of divisors, Amer. Math. Monthly, Vol. 68, No. 2 (Feb., 1961), pp. 120-124
(Mma) nn =10000; t = {}; Do[z = n + n*I; s = DivisorSigma[1, z]; If[Mod[s, z] == 0, AppendTo[t, n]; Print[{z, s, s/z}]], {n, nn}]; t
nonn,hard
T. D. Noe, Jul 12 2016