A332570 Numbers that are norm-abundant in Gaussian integers.

2, 4, 5, 6, 9, 10, 12, 13, 14, 15, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 58, 60, 62, 63, 64, 65, 66, 68, 70, 72, 74, 75, 76, 78, 80, 81, 82, 84, 85, 86, 87, 88, 90, 91, 92, 94, 95, 96

Offset

1,1

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 6, 70, 711, 7002, 69925, 701081, 7016287, 70074003, 700557394, 7007078826, ... Apparently this sequence has an asymptotic density of ~0.7.

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.

Example

2 is norm-abundant since sigma(2) = 2 + 3*i and N(2 + 3*i) = 2^2 + 3^2 = 13 > 2 * 2^2 = 8.

Mathematica

normAbQ[z_] := Abs[DivisorSigma[1, z, GaussianIntegers -> True]]^2 > 2*Abs[z]^2; Select[Range[100], normAbQ]

Crossrefs

Cf. A005101, A103228, A103229, A103230, A332320, A332321, A332571, A332572.

Sequence in context: A361936 A069470 A254820 * A047435 A331085 A132791

Adjacent sequences: A332567 A332568 A332569 * A332571 A332572 A332573

Keyword

nonn

Author

Amiram Eldar, Feb 16 2020

Status

approved