OFFSET
1,1
COMMENTS
For a Gaussian integer z, let the sum of the proper divisors be denoted by s(z) = sigma(z)-z, where sigma(z) is sum of the divisors of z, as defined by Spira for Gaussian integers. Then z is an amicable Gaussian number if z and s(z) are different and z = s(s(z)). The smallest Gaussian amicable number in the first quadrant is 8008+3960i.
LINKS
R. Spira, The Complex Sum Of Divisors, American Mathematical Monthly, 1961 Vol. 68, pp. 120-124.
Eric Weisstein's World of Mathematics, Amicable Pair
EXAMPLE
For z=-1105+1020i, we have s(z)=-2639-1228i and s(s(z))=z.
MATHEMATICA
s[z_Complex] := DivisorSigma[1, z]-z; nn=10000; lst={}; Do[d=a^2+b^2; If[d<nn^2, z=a+b*I; Do[If[s[s[z]]==z, AppendTo[lst, {d, z}]]; z=z*I, {4}]], {a, nn}, {b, nn}]; Re[Transpose[Sort[lst]][[2]]]
CROSSREFS
KEYWORD
sign,more
AUTHOR
T. D. Noe, Jan 19 2005
STATUS
approved