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 1-5 of 5 results.

A102531 Real part of absolute Gaussian perfect numbers, in order of increasing magnitude.

Original entry on oeis.org

3, 15, 6, 19, 111, 91, 159, 72, 472, 904, 2584, 1616, 999, 4328, 702, 4424, 7048, 7328, 2474, 9352, 7144, 7240, 5117, 739, 6327, 15128, 13168, 1263, 14280, 3224, 21704, 15160, 21992, 14044, 23132, 9135, 23656, 24614, 7272, 15464, 9040, 28424, 30956, 14728, 32399
Offset: 1

Views

Author

T. D. Noe, Jan 13 2005

Keywords

Comments

An absolute Gaussian perfect number z satisfies abs(sigma(z)-z) = abs(z), where sigma(z) is sum of the divisors of z, as defined by Spira for Gaussian integers.

Examples

			For z=3+7i, we have sigma(z)-z = 7+3i, which has the same magnitude as z.

Links

Crossrefs

See A102532 for the imaginary part.
Cf. A102506 and A102507 (Gaussian multiperfect numbers). See also A101366, A101367.

Programs

  • Mathematica
    lst={}; nn=1000; Do[z=a+b*I; If[Abs[z]<=nn && Abs[(DivisorSigma[1, z]-z)] == Abs[z], AppendTo[lst, {Abs[z]^2, z}]], {a, nn}, {b, nn}]; Re[Transpose[Sort[lst]][[2]]]

Extensions

a(22)-a(45) from Amiram Eldar, Feb 10 2020

A101366 Perfect Abs: Imaginary part of complex z such that Abs[(Total[Divisors[z]]-z)]=Abs[z].

Original entry on oeis.org

3, 7, 8, 42, 48, 57, 33, 82, 78, 77, 83, 189, 154, 92, 321, 341, 549, 664, 106, 1034, 2929, 4072, 5049, 3037, 6957, 7097, 5051
Offset: 0

Views

Author

Ed Pegg Jr, Jan 13 2005

Keywords

Comments

Having Perfect Abs is not as good as being Perfect. A complex number can also have Abundant Abs or Deficient Abs.

Examples

			The divisors for 269+92i are: 1, 2+I, 3+4i, 6+5i, 7+2i, 7+16i, 12+11i, 13+34i, 17+126i, 32+47i, 39+2i, 269+92i. The (sum - k) is 139+248i. Abs[139+248i] == Abs[269+92i]
The sequence of complex z's is: 5+3i, 3+7i, 19+8i, 15+42i, 6+57i, 29+48i, 19+82i, 74+33i, 111+78i, 147+77i, 185+83i, 91+189i, 197+154i, 269+92i, 122+321i, 159+341i, 72+549i, ...

Crossrefs

Cf. A101367, A102526, A102531, A102532.

Programs

  • Mathematica
    Im[Sort[Select[Flatten[Table[a + b I, {a, 1, 500}, {b, 1, 500}]], Abs[Total[Divisors[ # ]] - # ] == Abs[ # ] &], Abs[ #1] < Abs[ #2] &]]

Extensions

Ten more terms from Hans Havermann, Jan 15 2005

A101367 Perfect Abs. Real part of complex z such that Abs[(Total[Divisors[z]]-z)]=Abs[z].

Original entry on oeis.org

5, 3, 19, 15, 29, 6, 74, 19, 111, 147, 185, 91, 197, 269, 122, 159, 72, 827, 1487, 2903, 968, 999, 702, 5803, 326, 2474, 7871
Offset: 0

Views

Author

Ed Pegg Jr, Jan 13 2005

Keywords

Comments

Having Perfect Abs is not as good as being Perfect. A complex number can also have Abundant Abs or Deficient Abs.

Examples

			The divisors for 269+92i are: 1, 2+I, 3+4i, 6+5i, 7+2i, 7+16i, 12+11i, 13+34i, 17+126i, 32+47i, 39+2i, 269+92i. The (sum - k) is 139+248i. Abs[139+248i] == Abs[269+92i]

Crossrefs

Cf. A101366, A102527, A102531, A102532, A102506, A102507.

Programs

  • Mathematica
    Re[Sort[Select[Flatten[Table[a + b I, {a, 1, 500}, {b, 1, 500}]], Abs[Total[Divisors[ # ]] - # ] == Abs[ # ] &], Abs[ #1] < Abs[ #2] &]]

Extensions

Ten more terms from Hans Havermann, Jan 15 2005

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 *)
Showing 1-5 of 5 results.

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