A102531 - OEIS

%I #17 Feb 10 2020 06:13:44

%S 3,15,6,19,111,91,159,72,472,904,2584,1616,999,4328,702,4424,7048,

%T 7328,2474,9352,7144,7240,5117,739,6327,15128,13168,1263,14280,3224,

%U 21704,15160,21992,14044,23132,9135,23656,24614,7272,15464,9040,28424,30956,14728,32399

%N Real part of absolute Gaussian perfect numbers, in order of increasing magnitude.

%C An absolute Gaussian perfect number z satisfies abs(sigma(z)-z) = abs(z), where sigma(z) is sum of the divisors of z, as defined by Spira for Gaussian integers.

%H Amiram Eldar, <a href="/A102531/b102531.txt">Table of n, a(n) for n = 1..76</a>

%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.

%e For z=3+7i, we have sigma(z)-z = 7+3i, which has the same magnitude as z.

%t lst={}; nn=1000; Do[z=a+b*I; If[Abs[z]<=nn && Abs[(DivisorSigma[1, z]-z)] == Abs[z], AppendTo[lst, {Abs[z]^2, z}]], {a, nn}, {b, nn}]; Re[Transpose[Sort[lst]][[2]]]

%Y See A102532 for the imaginary part.

%Y Cf. A102506 and A102507 (Gaussian multiperfect numbers). See also A101366, A101367.

%K nonn

%O 1,1

%A _T. D. Noe_, Jan 13 2005

%E a(22)-a(45) from _Amiram Eldar_, Feb 10 2020