A332572 Numbers that are norm-deficient in Gaussian integers.

1, 3, 7, 8, 11, 16, 17, 19, 23, 29, 31, 37, 41, 43, 47, 49, 53, 57, 59, 61, 67, 69, 71, 73, 77, 79, 83, 89, 93, 97, 101, 103, 107, 109, 113, 121, 127, 128, 129, 131, 133, 137, 139, 141, 149, 151, 152, 157, 161, 163, 167, 173, 177, 179, 181, 184, 191, 193, 197

Offset: 1

Author

Amiram Eldar, Feb 16 2020

Keywords

• nonn

Comments

Numbers k such that N(sigma(k)) < 2*N(k) = 2*k^2, where 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.

The number of terms not exceeding 10^k for k = 1, 2, ... is 4, 30, 289, 2998, 30075, 298919, 2983713, 29925997, 299442606, 2992921174, ... Apparently this sequence has an asymptotic density of ~0.3.

Examples

			3 is norm-deficient since sigma(3) = 4 and N(4) = 4^2 = 16 < 2 * 3^2 = 18.
8 is norm-deficient since sigma(8) = -8 - 7*i and N(-8 - 7*i) = (-8)^2 + (-7)^2 = 113 < 2 * 8^2 = 128.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A005100, A103228, A103229, A103230, A332570.

Programs

Mathematica

normDefQ[z_] := Abs[DivisorSigma[1, z, GaussianIntegers -> True]]^2 < 2*Abs[z]^2; Select[Range[200], normDefQ]