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 1-4 of 4 results.

A066763 Numbers k such that (k, phi(k)) lies on a circle with integral radius centered at the origin, i.e., k^2 + phi(k)^2 is a square.

Original entry on oeis.org

15, 45, 70, 75, 135, 140, 225, 280, 350, 375, 377, 405, 490, 560, 627, 665, 675, 700, 980, 1120, 1125, 1215, 1400, 1750, 1875, 1881, 1960, 2025, 2240, 2450, 2800, 3325, 3375, 3430, 3500, 3645, 3655, 3920, 4480, 4655, 4900, 4901, 5600, 5625, 5643, 6075
Offset: 1

Views

Author

Joseph L. Pe, Jan 17 2002

Keywords

Examples

			15^2 + phi(15)^2 = 225 + 64 = 289 = 17^2, so 15 is a term of the sequence.

Links

Crossrefs

Cf. A066764, A066782.

Programs

  • Mathematica
    Select[ Range[ 1, 10^4 ], IntegerQ[ Sqrt[ #^2 + EulerPhi[ # ]^2 ] ] & ]
  • PARI
    isok(k) = { issquare(k^2 + eulerphi(k)^2) } \\ Harry J. Smith, Mar 23 2010

A066784 Numbers n such that (n, sigma(n)) lies on the hyperbola y^2 - x^2 = m^2, for some natural number m, i.e., sigma(n)^2 - n^2 = m^2.

Original entry on oeis.org

1, 90, 392448, 411264, 804384, 871416, 1284192, 1935360, 7456512, 396168192, 24193572480, 43171285248, 54585498240, 63178786944, 123570274464, 730078562304, 823442861592, 1420069242240, 2354025332736, 2506251331584, 3606011136000, 3697798293504, 9951618862080
Offset: 1

Views

Author

Joseph L. Pe, Jan 18 2002

Keywords

Examples

			sigma(90)^2 - 90^2 = 234^2 - 90^2 = 216^2, so 90 is a term of the sequence.

Crossrefs

Cf. A066764.

Programs

  • Mathematica
    Select[ Range[ 1, 10^6 ], IntegerQ[ Sqrt[ DivisorSigma[ 1, # ]^2 - #^2 ] ] & ]

Extensions

a(7)-a(10) from Sean A. Irvine, Nov 06 2023
a(11)-a(23) from Martin Ehrenstein, Jul 12 2024

A066785 Numbers k such that (k, phi(k), sigma(k)) lies on a sphere with integral radius centered at the origin, i.e., k^2 + phi(k)^2 + sigma(k)^2 is a square.

Original entry on oeis.org

603, 1414, 1444, 2093, 7434, 12822, 66584, 66960, 91000, 138645, 160130, 167400, 169212, 210476, 218592, 229026, 236572, 265977, 304128, 344790, 493722, 618570, 754110, 950158, 1091385, 1113480, 1760616, 1761409, 2116116, 2733131, 2776928
Offset: 1

Views

Author

Joseph L. Pe, Jan 18 2002

Keywords

Examples

			603^2 + phi(603)^2 + sigma(603)^2 = 603^2 + 396^2 + 884^2 = 1141^2, so 603 is a term of the sequence.

Links

Crossrefs

Cf. A066763, A066764.

Programs

  • Mathematica
    Select[ Range[ 1, 10^6 ], IntegerQ[ Sqrt[ #^2 + DivisorSigma[ 1, # ]^2 + EulerPhi[ # ]^2 ] ] & ]
  • PARI
    isok(k) = { issquare(k^2 + eulerphi(k)^2 + sigma(k)^2) } \\ Harry J. Smith, Mar 25 2010

Extensions

Terms a(25)-a(31) from Harry J. Smith, Mar 25 2010

A243455 Numbers n such that (n, sigma(n), tau(n)) lies on a sphere with integral radius centered at the origin, i.e., n^2 + sigma(n)^2 + tau(n)^2 is a square.

Original entry on oeis.org

6, 61, 2089, 3606, 18585, 28710, 70981, 121374, 176529, 520320, 970783, 62788800, 114682878, 144653952, 174635334, 182054895, 3857228169, 4349012190, 15994971740, 17587660602, 22842677823, 65183331200, 66928439760
Offset: 1

Views

Author

Michel Marcus, Jun 05 2014

Keywords

Comments

a(17) > 3*10^9. - Jinyuan Wang, Jul 09 2019
a(24) > 10^12, if it exists. - Giovanni Resta, Jul 17 2019

Examples

			6^2 + tau(6)^2 + sigma(6)^2 = 36 + 16 + 144 = 196 = 14^2. So 6 is in the sequence.

Crossrefs

Cf. A000005 (tau), A000203 (sigma), A066764 (analog in 2d).

Programs

  • PARI
    isok(n) = issquare(n^2 + numdiv(n)^2 + sigma(n)^2);

Extensions

a(13)-a(16) from Jinyuan Wang, Jul 09 2019
a(17)-a(23) from Giovanni Resta, Jul 17 2019
Showing 1-4 of 4 results.

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