A066763 -id:A066763 - cp's OEIS frontend

A066764 Numbers k such that (k, sigma(k)) lies on a circle with integral radius centered at the origin, i.e., k^2 + sigma(k)^2 is a square.

3, 8, 30, 140, 315, 319, 460, 840, 864, 936, 1040, 1287, 1836, 2480, 2600, 6076, 6200, 7788, 11398, 12035, 12415, 12584, 13260, 13734, 15886, 18480, 26078, 33820, 40640, 42665, 46200, 55860, 68352, 70266, 70645, 75330, 78939, 81740, 98230

Offset: 1

Joseph L. Pe, Jan 17 2002

nonn

Also numbers such that A017665(k)^2 + A017666(k)^2 is a square. - Michel Marcus, May 23 2015

			3^2 + sigma(3)^2 = 9 + 16 = 5^2; so 3 is a term of the sequence.

Harry J. Smith, Table of n, a(n) for n = 1..731

Cf. A017665, A017666, A066763.

Select[ Range[ 1, 10^5 ], IntegerQ[ Sqrt[ #^2 + DivisorSigma[ 1, # ]^2 ] ] & ]

isok(k) = { issquare(k^2 + sigma(k)^2) } \\ Harry J. Smith, Mar 24 2010

A066782 Numbers k such that (k, phi(k)) lies on the hyperbola x^2 - y^2 = m^2, for some natural number m, i.e., k^2 - phi(k)^2 is a square.

Original entry on oeis.org

1, 5, 13, 25, 34, 41, 61, 68, 113, 125, 136, 169, 181, 219, 222, 272, 313, 390, 421, 444, 482, 544, 578, 613, 625, 657, 666, 761, 780, 888, 964, 979, 1013, 1088, 1156, 1170, 1201, 1301, 1332, 1560, 1681, 1741, 1776, 1861

Offset: 1

Joseph L. Pe, Jan 18 2002

nonn

			5^2 - phi(5)^2 = 25 - 16 = 3^2, so 5 is a term of the sequence.

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A066763.

Select[ Range[ 1, 10^4 ], IntegerQ[ Sqrt[ #^2 - EulerPhi[ # ]^2 ] ] & ]

isok(k) = { issquare(k^2 - eulerphi(k)^2) } \\ Harry J. Smith, Mar 25 2010

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

Joseph L. Pe, Jan 18 2002

nonn

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

Harry J. Smith, Table of n, a(n) for n = 1..82

Cf. A066763, A066764.

Select[ Range[ 1, 10^6 ], IntegerQ[ Sqrt[ #^2 + DivisorSigma[ 1, # ]^2 + EulerPhi[ # ]^2 ] ] & ]

isok(k) = { issquare(k^2 + eulerphi(k)^2 + sigma(k)^2) } \\ Harry J. Smith, Mar 25 2010

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

cp's OEIS Frontend

A066764 Numbers k such that (k, sigma(k)) lies on a circle with integral radius centered at the origin, i.e., k^2 + sigma(k)^2 is a square.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A066782 Numbers k such that (k, phi(k)) lies on the hyperbola x^2 - y^2 = m^2, for some natural number m, i.e., k^2 - phi(k)^2 is a square.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

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.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions