A102924 - OEIS

%I #9 Feb 16 2025 08:32:56

%S -1105,-1895,-2639,-3235,-3433,-3970,-4694,-3549,-766,-4478,-6880,

%T 5356,-6468,8008,4232,-8547

%N Real part of Gaussian amicable numbers in order of increasing magnitude. See A102925 for the imaginary part.

%C 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.

%H R. Spira, <a href="http://www.jstor.org/stable/2312472">The Complex Sum Of Divisors</a>, American Mathematical Monthly, 1961 Vol. 68, pp. 120-124.

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/AmicablePair.html">Amicable Pair</a>

%e For z=-1105+1020i, we have s(z)=-2639-1228i and s(s(z))=z.

%t 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]]]

%Y Cf. A102506 (Gaussian multiperfect numbers), A102531 (absolute Gaussian perfect numbers).

%K sign,more,changed

%O 1,1

%A _T. D. Noe_, Jan 19 2005