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A332572
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Numbers that are norm-deficient in Gaussian integers.
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4
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1, 3, 7, 8, 11, 16, 17, 19, 23, 29, 31, 37, 41, 43, 47, 49, 53, 57, 59, 61, 67, 69, 71, 73, 77, 79, 83, 89, 93, 97, 101, 103, 107, 109, 113, 121, 127, 128, 129, 131, 133, 137, 139, 141, 149, 151, 152, 157, 161, 163, 167, 173, 177, 179, 181, 184, 191, 193, 197
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OFFSET
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1,2
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COMMENTS
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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 4, 30, 289, 2998, 30075, 298919, 2983713, 29925997, 299442606, 2992921174, ... Apparently this sequence has an asymptotic density of ~0.3.
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LINKS
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EXAMPLE
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3 is norm-deficient since sigma(3) = 4 and N(4) = 4^2 = 16 < 2 * 3^2 = 18.
8 is norm-deficient since sigma(8) = -8 - 7*i and N(-8 - 7*i) = (-8)^2 + (-7)^2 = 113 < 2 * 8^2 = 128.
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MATHEMATICA
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normDefQ[z_] := Abs[DivisorSigma[1, z, GaussianIntegers -> True]]^2 < 2*Abs[z]^2; Select[Range[200], normDefQ]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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