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.

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

A332472 The real part of the sum of unitary divisors function (usigma) generalized for Gaussian integers.

Original entry on oeis.org

1, 1, 4, -3, 4, 4, 8, 1, 10, -12, 12, -12, 6, 8, 16, 17, 6, 10, 20, -12, 32, 12, 24, 4, -24, -30, 28, -24, 8, -48, 32, 1, 48, -38, 32, -30, 8, 20, 24, 68, 10, 32, 44, -36, 40, 24, 48, 68, 50, -40, 24, -18, 10, 28, 48, 8, 80, -64, 60, -48, 12, 32, 80, -63, -120
Offset: 1

Views

Author

Amiram Eldar, Feb 13 2020

Keywords

Comments

If n = u * Product_{i} p_i^e_i, where u is a unit (1, i, -1 or -i), and p_i is a Gaussian prime with Re(p_i) > 0, then usigma(n) = Product_{i} (p_i^e_i + 1).
a(n) = A103228(n) for odd squarefree numbers (A056911), i.e., numbers n such that A318608(n) != 0.

Examples

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

Links

Crossrefs

Cf. A034448, A103228, A332473 (the imaginary 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_] := Re[usigma[n]]; Array[a, 100]

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]

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).

A332478 Number that are unitary norm-multiply-perfect numbers in Gaussian integers.

Original entry on oeis.org

1, 10, 12, 20160, 15713280, 137592000, 44289146880
Offset: 1

Views

Author

Amiram Eldar, Feb 13 2020

Keywords

Comments

Numbers k such that their norm of sum of unitary divisors in Gaussian integers, A332474(k), is divisible by their norm, k^2.
The corresponding ratios A332474(a(n))/(a(n)^2) are 1, 4, 1, 5, 5, 2, 5.

Examples

			10 is a term since its sum of unitary divisors in Gaussian integers is -12 + 16*i, whose norm (-12)^2 + 16^2 = 400 is divisible by 10^2 = 100.

Crossrefs

Cf. A002827, A327158, A332318, A332319, A332474.

Programs

  • Mathematica
    f[p_, e_] := If[Abs[p] == 1, 1, (p^e + 1)]; Select[Range[21000], Divisible[Abs[ Times @@ f @@@ FactorInteger[#, GaussianIntegers -> True]]^2, #^2] &]
Showing 1-5 of 5 results.

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

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