A332321 -id:A332321 - cp's OEIS frontend

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

Amiram Eldar, Feb 16 2020

nonn

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.

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

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.

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.

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

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

A332320 Numbers k that are highly norm-abundant in Gaussian integers, i.e., A103230(m) < A103230(k) for all m < k.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 10, 15, 18, 20, 26, 30, 50, 60, 70, 78, 90, 130, 150, 170, 180, 210, 260, 270, 330, 390, 510, 630, 780, 870, 910, 990, 1020, 1050, 1110, 1170, 1530, 1890, 1950, 2210, 2340, 2550, 2730, 3510, 4290, 4590, 5070, 5460, 5610, 5850, 6630, 8190, 10530

Offset: 1

Amiram Eldar, Feb 09 2020

nonn

Analogous to highly abundant numbers (A002093), with the norm of the sum of divisors function generalized for Gaussian integers (A103230) instead of the sum of divisors function (A000203).

			The first 6 terms of A103230 are 1, 13, 16, 41, 80, 208, 64, 113, 169, 1040. The record values occur at n = 1, 2, 3, 4, 5, 6, 10, the first 7 terms of this sequence.

Amiram Eldar, Table of n, a(n) for n = 1..375
Robert Spira, The Complex Sum of Divisors, The American Mathematical Monthly, Vol. 68, No. 2 (1961), pp. 120-124.

Cf. A000203, A002093, A103228, A103229, A103230, A279254, A332321.

s[n_] := Abs[DivisorSigma[1, n, GaussianIntegers -> True]]^2; sm = 0; seq = {}; Do[s1 = s[n]; If[s1 > sm, sm = s1; AppendTo[seq, n]], {n, 1,10^4}]; seq

cp's OEIS Frontend

A332570 Numbers that are norm-abundant in Gaussian integers.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica