A332571 Numbers that are primitive norm-abundant in Gaussian integers.

5, 9, 13, 21, 33, 119, 187, 203, 287, 543, 699, 807, 831, 843, 879, 939, 951, 1011, 1047, 1059, 1119, 1167, 1191, 1263, 1299, 1311, 1347, 1383, 1563, 1671, 1767, 1769, 1961, 2117, 2139, 2173, 2257, 2451, 2501, 2581, 2679, 2813, 2929, 2967, 2993, 3161, 3233, 3243

Offset: 1

Author

Amiram Eldar, Feb 16 2020

Keywords

• nonn

Comments

Numbers that are norm-abundant (A332570) in Gaussian integers and having no norm-abundant proper divisor.

Examples

			5 is primitive norm-abundant since it is norm-abundant, sigma(5) = 4 + 8*i and N(4 + 8*i) = 4^2 + 8^2 = 80 > 2 * 5^2 = 50, and none of the proper divisors of 5, {1, 1 + 2*i, 2 + i}, are norm-abundant: N(sigma(1)) = 1 < 2 * 1^2, N(sigma(1 + 2*i)) = N(2 + 2*i) = 8 < 2 * N(1 + 2*i) = 10, and N(sigma(2 + i)) = N(3 + i) = 10 = 2 * N(2 + i). (sigma(k) = A103228(k) + i*A103229(k) is the sum of divisors of k in Gaussian integers, i is the imaginary unit, and N(z) = Re(z)^2 + Im(z)^2 is the norm of the complex number z.)
		

References

• Miriam Hausman, On Norm Abundant Gaussian Integers, The Journal of the Indian Mathematical Society, Vol. 49 (1987), pp. 119-123.
• József Sándor, Dragoslav S. Mitrinovic and Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, chapter III, page 120.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Robert Spira, The Complex Sum of Divisors, The American Mathematical Monthly, Vol. 68, No. 2 (1961), pp. 120-124.

Crossrefs

Cf. A091191, A103228, A103229, A103230, A332570.

Programs

Mathematica

normAbQ[z_] := normAbQ[z] = Abs[DivisorSigma[1, z, GaussianIntegers -> True]]^2 > 2*Abs[z]^2; primNormAbQ[z_] := normAbQ[z] && !AnyTrue[Most[Divisors[z, GaussianIntegers -> True]], normAbQ]; Select[Range[1000], primNormAbQ]