cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A332575 Least start of a run of exactly n consecutive numbers that are norm-abundant in Gaussian integers (A332570).

Original entry on oeis.org

2, 9, 4, 12, 24, 185, 114, 1649, 692, 4977, 1412, 416345, 22624, 72233, 199892, 25262152, 1351880, 130824185, 16305324, 1688906313, 9412730, 10393378914, 721753400
Offset: 1

Views

Author

Amiram Eldar, Feb 16 2020

Keywords

Examples

			a(2) = 9 since 9 and 10 are the least pair of 2 consecutive numbers that are norm-abundant in Gaussian integers, and 8 and 11 are not norm-abundant.

Crossrefs

Cf. A005101, A096399, A096536, A103228, A103229, A103230, A332570.

Programs

  • Mathematica
    normAbQ[z_] := Abs[DivisorSigma[1, z, GaussianIntegers -> True]]^2 > 2*Abs[z]^2; n = 1; count = 0; max = 15; seq = Table[0, {max}]; While[count < max, n1 = n; If[normAbQ[n], While[normAbQ[++n1]]; d = n1 - n; If[d <= max && seq[[d]] == 0, count++; seq[[d]] = n]]; n = n1 + 1]; seq

A332572 Numbers that are norm-deficient in Gaussian integers.

Original entry on oeis.org

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
Offset: 1

Views

Author

Amiram Eldar, Feb 16 2020

Keywords

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.

Examples

			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.

Links

Crossrefs

Cf. A005100, A103228, A103229, A103230, A332570.

Programs

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

A332571 Numbers that are primitive norm-abundant in Gaussian integers.

Original entry on oeis.org

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
Offset: 1

Views

Author

Amiram Eldar, Feb 16 2020

Keywords

Comments

Numbers that are norm-abundant (A332570) in Gaussian integers and having no norm-abundant proper divisor.

Examples

			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.)

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

Crossrefs

Cf. A091191, A103228, A103229, A103230, A332570.

Programs

  • 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[1000], primNormAbQ]
Showing 1-3 of 3 results.

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