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.

A332477 Numbers k that are unitary harmonic in Gaussian integers: k * A332476(k) is divisible by A332472(k) + i*A332473(k) (where i is the imaginary unit).

Original entry on oeis.org

1, 5, 12, 50, 60, 84, 300, 420, 450, 756, 900, 1950, 3780, 7800, 9900, 33150, 49140, 54600, 100800, 132600, 265200, 491400, 928200, 1856400, 8353800, 8884200, 16707600, 52211250, 65995776, 78566400, 182739375, 183783600, 208845000, 280348992, 293046000, 329978880
Offset: 1

Views

Author

Amiram Eldar, Feb 13 2020

Keywords

Comments

Analogous to unitary harmonic numbers (A006086), with the number and sum of unitary divisors functions generalized for Gaussian integers (A332476, A332472 + i * A332473) instead of the number and sum of unitary divisors functions (A034444, A034448).

Examples

			5 is a term since 5 * A332476(5)/(A332472(5) + i*A332473(5)) = 5 * 4/(4 + 8*i) = 1 - 2*i is a Gaussian integer.

Links

Crossrefs

Cf. A034444, A034448, A006086, A332317, A332472, A332473, A332476.

Programs

  • Mathematica
    sigma[p_, e_] := If[Abs[p] == 1, 1, (p^e + 1)]; tau[p_, e_] := If[Abs[p] == 1, 1, 2]; unitaryHarmonicQ[n_] := Divisible[n * Times @@ tau @@@ (f = FactorInteger[n, GaussianIntegers -> True]), Times @@ sigma @@@ f]; Select[Range[10^5], unitaryHarmonicQ]

A332473 The imaginary part of the sum of unitary divisors function (usigma) generalized for Gaussian integers.

Original entry on oeis.org

0, 2, 0, 0, 8, 8, 0, -8, 0, 16, 0, 0, 18, 16, 32, 0, 22, 20, 0, -24, 0, 24, 0, -32, 8, 30, 0, 0, 36, 64, 0, 32, 0, 34, 64, 0, 44, 40, 72, -24, 50, 64, 0, 0, 80, 48, 0, 0, 0, -40, 88, -54, 62, 56, 96, -64, 0, 52, 0, -96, 72, 64, 0, 0, 120, 96, 0, -66, 0, 128, 0
Offset: 1

Views

Author

Amiram Eldar, Feb 13 2020

Keywords

Comments

See A332472 for a description.

Examples

			a(4) = 0 since 4 = -(1 + i)^4 in Gaussian integers (i is the imaginary unit), so usigma(4) = (1 + i)^4 + 1 = -3, and a(4) = Im(-3) = 0.

Links

Crossrefs

Cf. A034448, A103229, A332472 (the real part), A332474 (the norm).

Programs

  • Mathematica
    f[p_, e_] := If[Abs[p] == 1, 1, (p^e + 1)]; usigma[n_] := Times @@ f @@@ FactorInteger[n, GaussianIntegers -> True]; a[n_] := Im[usigma[n]]; Array[a, 100]

A332474 The norm of the sum of unitary divisors function (usigma) generalized for Gaussian integers.

Original entry on oeis.org

1, 5, 16, 9, 80, 80, 64, 65, 100, 400, 144, 144, 360, 320, 1280, 289, 520, 500, 400, 720, 1024, 720, 576, 1040, 640, 1800, 784, 576, 1360, 6400, 1024, 1025, 2304, 2600, 5120, 900, 2000, 2000, 5760, 5200, 2600, 5120, 1936, 1296, 8000, 2880, 2304, 4624, 2500, 3200
Offset: 1

Views

Author

Amiram Eldar, Feb 13 2020

Keywords

Comments

See A332472 for a description.

Examples

			a(4) = 9 since 4 = -(1 + i)^4 in Gaussian integers (i is the imaginary unit), so usigma(4) = (1 + i)^4 + 1 = -3, and a(4) = (-3)^2 + 0^2 = 9.

Links

Crossrefs

Cf. A034448, A103230, A332472 (the real part), A332473 (the imaginary part).

Programs

  • Mathematica
    f[p_, e_] := If[Abs[p] == 1, 1, (p^e + 1)]; usigma[n_] := Times @@ f @@@ FactorInteger[n, GaussianIntegers -> True]; a[n_] := Abs[usigma[n]]^2; Array[a, 100]

Formula

a(n) = A332472(n)^2 + A332473(n)^2.

A332476 The number of unitary divisors of n in Gaussian integers.

Original entry on oeis.org

1, 2, 2, 2, 4, 4, 2, 2, 2, 8, 2, 4, 4, 4, 8, 2, 4, 4, 2, 8, 4, 4, 2, 4, 4, 8, 2, 4, 4, 16, 2, 2, 4, 8, 8, 4, 4, 4, 8, 8, 4, 8, 2, 4, 8, 4, 2, 4, 2, 8, 8, 8, 4, 4, 8, 4, 4, 8, 2, 16, 4, 4, 4, 2, 16, 8, 2, 8, 4, 16, 2, 4, 4, 8, 8, 4, 4, 16, 2, 8, 2, 8, 2, 8, 16
Offset: 1

Views

Author

Amiram Eldar, Feb 13 2020

Keywords

Examples

			a(2) = 2 since 2 = -i * (1 + i)^2, so it has 2 unitary divisors (up to associates): 1 and (1 + i)^2.

Links

Crossrefs

Cf. A034444, A062327, A086275, A332472, A332473, A332474.

Programs

  • Mathematica
    f[p_, e_] := If[Abs[p] == 1, 1, 2]; a[n_] := Times @@ f @@@ FactorInteger[n, GaussianIntegers -> True]; Array[a, 100]

Formula

Multiplicative with a(p^e) = 4 if p == 1 (mod 4) and 2 otherwise.
a(n) = 2^A086275(n).

A332475 Numbers k such that k and k + 1 have the same norm of the sum of unitary divisors in Gaussian integers (A332474).

Original entry on oeis.org

5, 11, 37, 1738, 2772, 6600, 42251, 49913, 57816, 104754, 220324, 288350, 364452, 792156, 1711932, 1971475, 2607049, 2793473, 3211933, 3521148, 3526312, 4012736, 5805149, 5918276, 6522320, 6542147, 6635436, 7612267, 12604600, 14844791, 17078848, 19024332, 21177516
Offset: 1

Views

Author

Amiram Eldar, Feb 13 2020

Keywords

Examples

			5 is a term since A332474(5) = A332474(6) = 80.

Links

Crossrefs

Cf. A002961, A064125, A293183, A306985, A332315, A332472, A332473, A332474.

Programs

  • Mathematica
    f[p_, e_] := If[Abs[p] == 1, 1, (p^e + 1)]; normUsigma[n_] := Abs[Times @@ f @@@ FactorInteger[n, GaussianIntegers -> True]]^2; seq = {}; u1 = normUsigma[1]; Do[u2 = normUsigma[n]; If[u1 == u2, AppendTo[seq, n - 1]]; u1 = u2, {n, 2, 10^6}]; seq
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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