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 A332571 Numbers that are primitive norm-abundant in Gaussian integers. 2
 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 (list; graph; refs; listen; history; text; internal format)
 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, 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

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Last modified September 22 00:10 EDT 2023. Contains 365503 sequences. (Running on oeis4.)