A332572 -id:A332572 - cp's OEIS frontend

A332573 Numbers k such that k and k + 1 are both norm-deficient in Gaussian integers (A332572).

7, 16, 127, 128, 151, 248, 256, 343, 472, 536, 568, 631, 632, 751, 752, 823, 856, 943, 1048, 1111, 1136, 1207, 1303, 1327, 1328, 1336, 1432, 1527, 1528, 1591, 1687, 1688, 1711, 1712, 1783, 1816, 1912, 2031, 2032, 2047, 2048, 2103, 2167, 2263, 2416, 2487, 2488

Offset: 1

Amiram Eldar, Feb 16 2020

nonn

			7 is a term since both 7 and 7 + 1 = 8 are norm-deficient in Gaussian integers (A332572).

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

Cf. A005100, A103228, A103229, A103230, A317047, A332572.

normDefQ[z_] := Abs[DivisorSigma[1, z, GaussianIntegers -> True]]^2 < 2*Abs[z]^2; Select[Range[2500], normDefQ[#] && normDefQ[# + 1] &]

A332574 Numbers k such that k, k + 1 and k + 2 are all norm-deficient in Gaussian integers (A332572).

Original entry on oeis.org

127, 631, 751, 1327, 1527, 1687, 1711, 2031, 2047, 2487, 2647, 3207, 3271, 3351, 3511, 3831, 4567, 4791, 4911, 5127, 6007, 6087, 6711, 7431, 8247, 8367, 8391, 8407, 8551, 8751, 8871, 9031, 9447, 9991, 10407, 10551, 10887, 10927, 11631, 12471, 12567, 12631, 13807

Offset: 1

Amiram Eldar, Feb 16 2020

nonn

			127 is a term since 127, 128 and 129 are all norm-deficient in Gaussian integers (A332572).

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

Cf. A005100, A103228, A103229, A103230, A317047, A332572, A332573.

normDefQ[z_] := Abs[DivisorSigma[1, z, GaussianIntegers -> True]]^2 < 2*Abs[z]^2; Select[Range[10^4], AllTrue[# + Range[0, 2], normDefQ] &]

A332570 Numbers that are norm-abundant in Gaussian integers.

Original entry on oeis.org

2, 4, 5, 6, 9, 10, 12, 13, 14, 15, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 58, 60, 62, 63, 64, 65, 66, 68, 70, 72, 74, 75, 76, 78, 80, 81, 82, 84, 85, 86, 87, 88, 90, 91, 92, 94, 95, 96

Offset: 1

Amiram Eldar, Feb 16 2020

nonn

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 6, 70, 711, 7002, 69925, 701081, 7016287, 70074003, 700557394, 7007078826, ... Apparently this sequence has an asymptotic density of ~0.7.

			2 is norm-abundant since sigma(2) = 2 + 3*i and N(2 + 3*i) = 2^2 + 3^2 = 13 > 2 * 2^2 = 8.

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.

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.

Cf. A005101, A103228, A103229, A103230, A332320, A332321, A332571, A332572.

normAbQ[z_] := Abs[DivisorSigma[1, z, GaussianIntegers -> True]]^2 > 2*Abs[z]^2; Select[Range[100], normAbQ]

cp's OEIS Frontend

A332573 Numbers k such that k and k + 1 are both norm-deficient in Gaussian integers (A332572).

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A332574 Numbers k such that k, k + 1 and k + 2 are all norm-deficient in Gaussian integers (A332572).

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Mathematica

A332570 Numbers that are norm-abundant in Gaussian integers.

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