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 1-6 of 6 results.

A100889 Real part of the multiplier m of the multiperfect Gaussian integer A100884(n)+i*A100885(n).

Original entry on oeis.org

1, 2, -1, 2, -1, 2, -2, 1, -1, 0, -3, 1, -3, 0, -2, -4, -3, 1, -4, 2, 0, 3, 0, 1, 3, 3, 4, 3, 0, -1, 0, 5, 3, -5, 4, 0, 4, 0, 0, 0, -4
Offset: 1

Views

Author

Yasutoshi Kohmoto

Keywords

Comments

Let the Gaussian integer z = A100884(n)+i*A100885(n). Then a(n) = Re(m) = Re( sigma(z)/z) .

Crossrefs

Cf. A100884, A100885, A100890.

Extensions

Edited, all values replaced by R. J. Mathar, Mar 12 2010
a(10)-a(41) from Amiram Eldar, Feb 10 2020

A100884 Real parts of multiperfect Gaussian numbers z, sorted with respect to |z| and Re(z).

Original entry on oeis.org

1, 1, 6, 5, 10, 12, 60, 72, 28, 100, 108, 120, 204, 300, 263, 140, 526, 912, 150, 720, 1470, 1520, 1200, 1704, 672, 600, 4560, 4828, 3600, 5584, 5880, 4680, 6312, 6240, 1800, 2160, 14484, 17640, 8984, 72824, 62400
Offset: 1

Views

Author

Yasutoshi Kohmoto

Keywords

Comments

Sort the Gaussian integers z in the first quadrant according to increasing modulus |z|, and within the same modulus according to increasing Re(z): 1, 1+i, 2, 1+2i, 2+i, 2+2i, 3, 1+3i, 3+i, 2+3i, 3+2i,...
If z divides the value of sigma(z), defined in A103228, i.e., if sigma(z)=z*m with m some Gaussian integer (m not necessarily in the first quadrant), add Re(z) to the sequence.

Examples

			For z = 1, sigma(z) = 1 and m = sigma(z)/z = 1, which adds 1 to the sequence.
For z = 1+3i, sigma(z) = 5+5i and m = sigma(z)/z = 2-i, which adds 1 to the sequence.
For z = 6+2i, sigma(z) = -10+10i and m = sigma(z)/z = -1+2i, which adds 6 to the sequence.
For z = 5+5i, sigma(z) = 20i and m = sigma(z)/ z= 2+2i, which adds 5 to the sequence.
For z = (1+i)^7 = 8-8i, the divisors are 1, 1+i, (1+i)^2 = 2i, (1+i)^3 = -2+2i, (1+i)^4 = -4, (1+i)^5= -4-4i, (1+i)^6 = -8i, (1+i)^7 = 8-8i. So sigma(z) is 1 +1+i +2i -2+2i -4 -4-4i -8i +8-8i = -15i and sigma(z)/z is m = -15i/(8-8i) which is not a Gaussian integer, so Re(z)=8 is NOT added to the sequence.

Crossrefs

Cf. A100885, A100889, A100890.

Extensions

Entirely rewritten, including the a(n), by R. J. Mathar, Mar 12 2010
a(10)-a(41) from Amiram Eldar, Feb 10 2020

A100890 Imaginary part of multiplier m of the Gaussian multiperfect number associated with the real part A100889(n).

Original entry on oeis.org

0, -1, 2, 2, 3, 2, -1, -2, 2, -3, -1, -3, -1, -4, 2, 0, -1, -2, 0, 1, -4, 0, 3, -3, 1, 0, 0, 0, 4, 3, 3, 0, 1, 0, 0, 4, 0, 4, -4, 3, -1
Offset: 1

Views

Author

Yasutoshi Kohmoto

Keywords

Comments

Let the Gaussian integer z = A100884(n)+i*A100885(n). Then a(n) = Im(m) = Im( sigma(z)/z).

Crossrefs

Cf. A100884, A100885, A100889.

Extensions

Edited, all values replaced by R. J. Mathar, Mar 12 2010
a(10)-a(41) from Amiram Eldar, Feb 10 2020

A332318 Real part of Gaussian norm-multiply-perfect numbers, in order of increasing norm.

Original entry on oeis.org

1, 2, 1, 3, 6, 5, 6, 10, 2, 8, 16, 12, 21, 18, 26, 30, 13, 43, 28, 6, 52, 60, 20, 10, 72, 56, 26, 28, 85, 95, 20, 100, 58, 80, 40, 36, 108, 120, 48, 190, 144, 176, 204, 38, 106, 214, 4, 84, 232, 276, 136, 216, 300, 174, 181, 263, 216, 312, 340, 140, 464, 20, 380
Offset: 1

Views

Author

Amiram Eldar, Feb 09 2020

Keywords

Comments

Norm-multiply-perfect numbers where defined by Spira as Gaussian integers z such that norm(sigma(z)) is divisible by norm(z), where sigma(z) is the sum of the divisors of z = a + b*i and norm(z) = z * conj(z) = a^2 + b^2.
If the ratio norm(sigma(z))/norm(z) is 2 then the number is called a norm-perfect. The norm-perfect numbers correspond to n = 1, 13, 26, ... They are 2 + i, 21 + 22*i, 56 + 64*i, ...
Only nonnegative terms are included, since if z = a + b*i is a norm-multiply-perfect number then z*i, -z and -z*i are also norm-multiply-perfect numbers and one of the four has nonnegative a and b. Terms with the same norm are ordered by their real parts.
The corresponding imaginary parts are in A332319.
The corresponding norms are 1, 5, 10, 10, 40, 50, 52, 200, 260, 260, 260, 468, ...
The corresponding ratios norm(sigma(k))/k are 1, 2, 5, 4, 5, 8, 5, 10, 9, 10, ...

Examples

			2 + i is a norm-perfect number since sigma(2 + i) = 3 + i, and (3^2 + 1^2) = 10 = 2 * 5 = 2 * (2^1 + 1^2).
1 + 3*i is a norm-multiply-perfect number since sigma(1 + 3*i) = 5 + 5*i, and (5^2 + 5^2) = 50 = 5 * 10 = 5 * (1^2 + 3^2).

Links

Crossrefs

Cf. A000396, A007691, A100884, A100885, A100889, A100890, A102506, A102507, A102531, A102532, A103228, A103229, A103230, A332319.

Programs

  • Mathematica
    csigma[z_] := DivisorSigma[1, z, GaussianIntegers -> True]; normultPerf[z_] := Divisible[Abs[csigma[z]]^2, Abs[z]^2]; seq = {}; max = 10^2; Do[z = a + b*I; If[Abs[z] <= max && normultPerf[z], AppendTo[seq, {Abs[z]^2, z}]], {a, 1, max}, {b, 0, max}]; Re[Transpose[Sort[seq]][[2]]] (* after T. D. Noe at A102531 *)

A332319 Imaginary part of Gaussian norm-multiply-perfect numbers, in order of increasing norm.

Original entry on oeis.org

0, 1, 3, 1, 2, 5, 4, 10, 16, 14, 2, 18, 22, 26, 18, 20, 41, 1, 36, 48, 24, 12, 65, 70, 24, 64, 82, 88, 45, 15, 100, 20, 94, 100, 130, 132, 84, 120, 184, 30, 148, 108, 32, 216, 192, 48, 228, 212, 51, 24, 252, 188, 60, 282, 283, 209, 312, 216, 198, 440, 102, 490
Offset: 1

Views

Author

Amiram Eldar, Feb 09 2020

Keywords

Comments

See A332318 (the corresponding real parts) for the definition of Gaussian norm-multiply-perfect numbers.

Links

Crossrefs

Cf. A000396, A007691, A100884, A100885, A100889, A100890, A102506, A102507, A102531, A102532, A103228, A103229, A103230, A332318.

Programs

  • Mathematica
    csigma[z_] := DivisorSigma[1, z, GaussianIntegers -> True]; normultPerf[z_] := Divisible[Abs[csigma[z]]^2, Abs[z]^2]; seq = {}; max = 10^2; Do[z = a + b*I; If[Abs[z] <= max && normultPerf[z], AppendTo[seq, {Abs[z]^2, z}]], {a, 1, max}, {b, 0, max}]; Im[Transpose[Sort[seq]][[2]]] (* after T. D. Noe at A102532 *)

A100888 Expansion of (3+x-x^2)/((1+x+x^2)(1-x-x^2)).

Original entry on oeis.org

3, 1, 2, 7, 7, 12, 23, 33, 54, 91, 143, 232, 379, 609, 986, 1599, 2583, 4180, 6767, 10945, 17710, 28659, 46367, 75024, 121395, 196417, 317810, 514231, 832039, 1346268, 2178311, 3524577, 5702886, 9227467, 14930351, 24157816, 39088171, 63245985
Offset: 0

Views

Author

Creighton Dement, Nov 21 2004

Keywords

Comments

This sequence was investigated in cooperation with Paul Barry. Generating floretion: - 0.5'i - 0.5'k - 0.5j' - 0.5'ii' + 0.5'jj' - 0.5'kk' + 0.5'ik' - 0.5'ki' ("jes"). A100885(n) = (1/2)(A100886(n) + A100887(n) - a(n)).

Crossrefs

Cf. A100885, A100886, A100887, A100889, A100890.

Programs

  • Mathematica
    a[0] = 3; a[1] = 1; a[2] = 2; a[3] = 7; a[n_] := a[n] = a[n - 2] + 2a[n - 3] + a[n - 4]; Table[ a[n], {n, 0, 37}] (* Robert G. Wilson v, Nov 26 2004 *)
CoefficientList[ Series[(3 + x - x^2)/((1 + x + x^2)(1 - x - x^2)), {x, 0, 37}], x] (* Robert G. Wilson v, Nov 26 2004 *)
  • PARI
    Vec((3+x-x^2)/((1+x+x^2)*(1-x-x^2))+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

Formula

a(n) = Fib(n+2) + sqrt(3)cos(2Pi*n/3 + Pi/6) + sin(2Pi*n/3 + Pi/6); a(n) = a(n-2) + 2a(n-3) + a(n-4), a(0) = 3, a(1) = 1, a(2) = 2, a(3) = 7.

Extensions

More terms from Robert G. Wilson v, Nov 26 2004
Showing 1-6 of 6 results.

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