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

1,1

Comments

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

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

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.)

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]

Crossrefs

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

Sequence in context: A166049 A334214 A053411 * A023382 A314793 A314794

Adjacent sequences: A332568 A332569 A332570 * A332572 A332573 A332574

Keyword

nonn

Author

Amiram Eldar, Feb 16 2020

Status

approved