

A332572


Numbers that are normdeficient in Gaussian integers.


4



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

1,2


COMMENTS

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.


LINKS

Amiram Eldar, Table of n, a(n) for n = 1..10000


EXAMPLE

3 is normdeficient since sigma(3) = 4 and N(4) = 4^2 = 16 < 2 * 3^2 = 18.
8 is normdeficient since sigma(8) = 8  7*i and N(8  7*i) = (8)^2 + (7)^2 = 113 < 2 * 8^2 = 128.


MATHEMATICA

normDefQ[z_] := Abs[DivisorSigma[1, z, GaussianIntegers > True]]^2 < 2*Abs[z]^2; Select[Range[200], normDefQ]


CROSSREFS

Cf. A005100, A103228, A103229, A103230, A332570.
Sequence in context: A278519 A007970 A255342 * A134258 A028972 A153030
Adjacent sequences: A332569 A332570 A332571 * A332573 A332574 A332575


KEYWORD

nonn


AUTHOR

Amiram Eldar, Feb 16 2020


STATUS

approved



