## S000947

First-quadrant complex numbers x + y*I, sorted by norm, that are multiply-perfect.

1, 3, 6, 2, 5, 5, 10, 10, 12, 18, 60, 12, 72, 24, 28, 88, 100, 20, 108, 84, 120, 120, 204, 32, 300, 60, 263, 209, 140, 440, 526, 418, 912, 120, 150, 950, 720, 768, 1470, 310, 1520, 200, 1200, 1280, 1704, 1152, 672, 2256, 600, 3800, 4560, 600, 4828, 404, 3600, 3840

1

multiply-perfect complex number x+y*I is a number whose divisors sum to an complex multiple of x+y*I. For instance, 1+3*I has divisors 1, 1+I, 1+3*I, and 2+I, which sum to (1+3*I)(2-I). Numbers in other quadrants can be formed by multiplying by -1, I, and -I. The plot above shows the numbers plotted in the complex plane. Unlike integer perfect numbers, which factor into 2^k * p, where p is prime, complex multiply-perfect numbers do not have such a nice factorization.

T. D. Noe, Plot of 41 complex numbers

T. D. Noe, Table of 82 terms

Eric W. Weisstein, Perfect Number

Eric W. Weisstein, Multiperfect Number

(Mma) nn = 1000; tc = {}; Do[z = n + k*I; s = DivisorSigma[1, z]; If[k != 0 && Mod[s, z] == 0, AppendTo[tc, z]; Print[{z, s, s/z}]], {n, nn}, {k, -n, n}]; tc = Select[Union[tc, -tc, I*tc, -I*tc], Re[#] > 0 && Im[#] > 0 &]; Sort[tc, Norm[#1] < Norm[#2] &]

Cf. A000396 (perfect numbers), S000948, S000949.

nonn,hard,nice

T. D. Noe, Oct 04 2016