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%I #7 Feb 17 2020 09:45:19
%S 1,3,7,8,11,16,17,19,23,29,31,37,41,43,47,49,53,57,59,61,67,69,71,73,
%T 77,79,83,89,93,97,101,103,107,109,113,121,127,128,129,131,133,137,
%U 139,141,149,151,152,157,161,163,167,173,177,179,181,184,191,193,197
%N Numbers that are norm-deficient in Gaussian integers.
%C 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.
%C 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.
%H Amiram Eldar, <a href="/A332572/b332572.txt">Table of n, a(n) for n = 1..10000</a>
%e 3 is norm-deficient since sigma(3) = 4 and N(4) = 4^2 = 16 < 2 * 3^2 = 18.
%e 8 is norm-deficient since sigma(8) = -8 - 7*i and N(-8 - 7*i) = (-8)^2 + (-7)^2 = 113 < 2 * 8^2 = 128.
%t normDefQ[z_] := Abs[DivisorSigma[1, z, GaussianIntegers -> True]]^2 < 2*Abs[z]^2; Select[Range[200], normDefQ]
%Y Cf. A005100, A103228, A103229, A103230, A332570.
%K nonn
%O 1,2
%A _Amiram Eldar_, Feb 16 2020
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