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

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, 26, 28, 85, 95, 20, 100, 58, 80, 40, 36, 108, 120, 48, 190, 144, 176, 204, 38, 106, 214, 4, 84, 232, 276, 136, 216, 300, 174, 181, 263, 216, 312, 340, 140, 464, 20, 380

Offset: 1

Author

Amiram Eldar, Feb 09 2020

Keywords

• nonn

Comments

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.

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, ...

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.

The corresponding imaginary parts are in A332319.

The corresponding norms are 1, 5, 10, 10, 40, 50, 52, 200, 260, 260, 260, 468, ...

The corresponding ratios norm(sigma(k))/k are 1, 2, 5, 4, 5, 8, 5, 10, 9, 10, ...

Examples

			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).
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).
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..182
• Wayne L. McDaniel, Perfect Gaussian integers, Acta Arithmetica, Vol. 2, No. 25 (1974), pp. 137-144.
• Robert Spira, The Complex Sum of Divisors, The American Mathematical Monthly, Vol. 68, No. 2 (1961), pp. 120-124.

Crossrefs

Cf. A000396, A007691, A100884, A100885, A100889, A100890, A102506, A102507, A102531, A102532, A103228, A103229, A103230, A332319.

Programs

Mathematica

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 *)