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A332318 Real part of Gaussian norm-multiply-perfect numbers, in order of increasing norm.

%I A332318 #17 Feb 11 2020 03:54:30
%S A332318 1,2,1,3,6,5,6,10,2,8,16,12,21,18,26,30,13,43,28,6,52,60,20,10,72,56,
%T A332318 26,28,85,95,20,100,58,80,40,36,108,120,48,190,144,176,204,38,106,214,
%U A332318 4,84,232,276,136,216,300,174,181,263,216,312,340,140,464,20,380
%N A332318 Real part of Gaussian norm-multiply-perfect numbers, in order of increasing norm.
%C A332318 Norm-multiply-perfect numbers where defined by Spira as Gaussian integers z such that norm(sigma(z)) is divisible by norm(z), where sigma(z) is the sum of the divisors of z = a + b*i and norm(z) = z * conj(z) = a^2 + b^2.
%C A332318 If the ratio norm(sigma(z))/norm(z) is 2 then the number is called a norm-perfect. The norm-perfect numbers correspond to n = 1, 13, 26, ... They are 2 + i, 21 + 22*i, 56 + 64*i, ...
%C A332318 Only nonnegative terms are included, since if z = a + b*i is a norm-multiply-perfect number then z*i, -z and -z*i are also norm-multiply-perfect numbers and one of the four has nonnegative a and b. Terms with the same norm are ordered by their real parts.
%C A332318 The corresponding imaginary parts are in A332319.
%C A332318 The corresponding norms are 1, 5, 10, 10, 40, 50, 52, 200, 260, 260, 260, 468, ...
%C A332318 The corresponding ratios norm(sigma(k))/k are 1, 2, 5, 4, 5, 8, 5, 10, 9, 10, ...
%H A332318 Amiram Eldar, <a href="/A332318/b332318.txt">Table of n, a(n) for n = 1..182</a>
%H A332318 Wayne L. McDaniel, <a href="https://eudml.org/doc/205261">Perfect Gaussian integers</a>, Acta Arithmetica, Vol. 2, No. 25 (1974), pp. 137-144.
%H A332318 Robert Spira, <a href="https://doi.org/10.1080/00029890.1961.11989634">The Complex Sum of Divisors</a>, The American Mathematical Monthly, Vol. 68, No. 2 (1961), pp. 120-124.
%e A332318 2 + i is a norm-perfect number since sigma(2 + i) = 3 + i, and (3^2 + 1^2) = 10 = 2 * 5 = 2 * (2^1 + 1^2).
%e A332318 1 + 3*i is a norm-multiply-perfect number since sigma(1 + 3*i) = 5 + 5*i, and (5^2 + 5^2) = 50 = 5 * 10 = 5 * (1^2 + 3^2).
%t A332318 csigma[z_] := DivisorSigma[1, z, GaussianIntegers -> True]; normultPerf[z_] := Divisible[Abs[csigma[z]]^2, Abs[z]^2]; seq = {}; max = 10^2; Do[z = a + b*I; If[Abs[z] <= max && normultPerf[z], AppendTo[seq, {Abs[z]^2, z}]], {a, 1, max}, {b, 0, max}]; Re[Transpose[Sort[seq]][[2]]] (* after _T. D. Noe_ at A102531 *)
%Y A332318 Cf. A000396, A007691, A100884, A100885, A100889, A100890, A102506, A102507, A102531, A102532, A103228, A103229, A103230, A332319.
%K A332318 nonn
%O A332318 1,2
%A A332318 _Amiram Eldar_, Feb 09 2020