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 1-6 of 6 results.

A238230 Numbers m such that if x = sigma(m)-phi(m)-tau(m)-m then m = sigma(x)-phi(x)-tau(x)-x.

Original entry on oeis.org

198, 294, 16008, 22232, 150030, 195320, 200274, 3038720, 12190720, 124904790, 167179722, 347943288, 426853240, 528995656, 646186568, 3588779502, 4798752860, 5376246738, 5898361924, 158380893880, 189740533470, 196271084296, 240458641570, 375653406648
Offset: 1

Views

Author

Paolo P. Lava, Feb 20 2014

Keywords

Comments

All numbers of the form 2^k*5*p, where p = (7*2^k-2*k-5)/3 is prime, are fixed points and thus terms. This happens for k = 9, 10, 124, 352, 1468, 3339, 4365,... - Giovanni Resta, Mar 26 2014
The fixed points (terms with x = m) are 198, 294, 195320, 3038720, 12190720, ... - Amiram Eldar, Mar 31 2019
a(20) > 10^11. - Giovanni Resta, Apr 04 2019

Examples

			Fixed points: 198, 294, 195320,...
sigma(16008) = 43200, phi(16008) = 4928, tau(16008) = 32 and 43200 - 4928 - 32- 16008 = 22232.
sigma(22232) = 47760, phi(22232) = 9504, tau(22232) = 16 and 47760 - 9504 - 16 - 22232 = 16008.

Crossrefs

Cf. A000005, A000010, A000203, A055971, A238225-A238229.

Programs

  • Maple
    with(numtheory); P:=proc(q)local a,n;
for n from 1 to q do a:=sigma(n)-phi(n)-tau(n)-n;
if a>0 and sigma(a)-phi(a)-tau(a)-a=n then print(n);
fi; od; end: P(10^6);
  • Mathematica
    f[n_] := If[n > 0, DivisorSigma[1, n] - EulerPhi[n] - DivisorSigma[0, n] - n, 0]; s={}; Do[ If[f[f[n]] == n, AppendTo[s, n]], {n, 1, 200000}]; s (* Amiram Eldar, Mar 31 2019 *)

Extensions

a(8)-a(15) from Giovanni Resta, Mar 26 2014
a(16)-a(19) from Amiram Eldar, Mar 31 2019
a(19) corrected by Kevin P. Thompson, Jan 12 2022
a(20)-a(23) from Kevin P. Thompson, Apr 17 2022
a(24) from Kevin P. Thompson, Jun 13 2022

A238226 Numbers k such that if x = sigma(k) + tau(k) - k then k = sigma(x) + tau(x) - x.

Original entry on oeis.org

1, 3, 14, 52, 130, 144, 184, 274, 300, 586, 656, 8648, 10434, 11470, 12008, 15774, 17034, 18802, 19270, 21032, 22088, 22184, 23288, 34688, 35394, 36872, 38744, 39790, 65324, 65392, 67628, 68476, 153868, 163676, 188468, 198628, 254526, 263890, 379026, 463390
Offset: 1

Views

Author

Paolo P. Lava, Feb 20 2014

Keywords

Comments

A083874 is a subset of this sequence: it lists the fixed points of the transform n -> sigma(n)+tau(n)-n.

Examples

			Fixed points: 1, 3, 14, 52, 130, 184, 656, 8648, 12008, 34688, ...
sigma(144) = 403, tau(144) = 15 and 403 + 15 - 144 = 274.
sigma(274) = 414, tau(274) = 4 and 414 + 4 - 274 = 144.

Links

Crossrefs

Cf. A000005, A000203, A083874, A238225, A238227-A238230.

Programs

  • Maple
    with(numtheory); P:=proc(q)local a,n;
for n from 1 to q do a:=sigma(n)+tau(n)-n;
if sigma(a)+tau(a)-a=n then print(n);
fi; od; end: P(10^6);
  • Mathematica
    f[n_] := DivisorSigma[0, n] + DivisorSigma[1, n] - n; s={}; Do[m = f[n]; If[f[m] == n, AppendTo[s, n]], {n, 1, 500000}]; s (* Amiram Eldar, Jul 12 2019 *)

A238227 Numbers n such that if x=sigma(n)-tau(n)-n then n=sigma(x)-tau(x)-x.

Original entry on oeis.org

1, 56, 66, 70, 992, 1012, 2260, 2516, 6042, 6902, 7192, 7210, 7232, 7750, 7912, 8178, 9086, 10792, 12198, 13706, 17272, 30592, 32778, 33352, 35032, 40166, 44034, 45010, 46670, 47710, 55374, 62296, 63688, 65570, 114256, 132916, 133892, 138244, 141236, 146804, 155572
Offset: 1

Views

Author

Paolo P. Lava, Feb 20 2014

Keywords

Comments

If the second term (4) is not considered, A056075 is almost a subset of this sequence: it lists the fixed points of the transform n -> sigma(n)-tau(n)-n.

Examples

			Fixed points: 56, 7192, 7232, 7912, 10792, ...
sigma(66) = 144, tau(66) = 8 and 144 - 8 - 66 = 70.
sigma(70) = 144, tau(70) = 8 and 144 - 8 - 70 = 66.

Links

Crossrefs

Cf. A000005, A000203, A056075, A238225, A238226, A238228, A238229, A238230.

Programs

  • Maple
    with(numtheory); P:=proc(q)local a,n;
for n from 1 to q do a:=sigma(n)-tau(n)-n;
if sigma(a)-tau(a)-a=n then print(n);
fi; od; end: P(10^6);

A238229 Numbers n such that if x = sigma(n)-phi(n)+tau(n)-n then n = sigma(x)-phi(x)+tau(x)-x.

Original entry on oeis.org

2, 4, 20, 66, 342, 34092, 40842, 41922, 46242, 55422, 207624, 259448, 533172, 547300, 571992, 667408, 1531032, 1786288, 10983114, 114013752, 133506680, 323277822, 347360860, 386144360, 387415458, 459603716, 476991704, 1443279992, 1539484232, 15537978552
Offset: 1

Views

Author

Paolo P. Lava, Feb 20 2014

Keywords

Comments

The fixed points (terms with x = n) are 2, 4, 20, 66, 342, 41922, 10983114, ... - Amiram Eldar, Mar 31 2019

Examples

			Fixed points: 2, 4, 20, 66, 342, 41922, ...
Amicable pairs: (34092, 40842), (46242, 55422), (207624, 259448), ...
sigma(34092) = 86268, phi(34092) = 11352, tau(34092) = 18 and 86268 - 11352 + 18 - 34092 = 40842.
sigma(40842) = 88530, phi(40842) = 13608, tau(40842) = 12 and 88530 - 13608 + 12 - 40842 = 34092.

Links

Crossrefs

Cf. A000005, A000010, A000203.
Cf. A238225, A238226, A238227, A238228, A238230.

Programs

  • Maple
    with(numtheory); P:=proc(q)local a,n;
for n from 1 to q do a:=sigma(n)-phi(n)+tau(n)-n;
if a>0 and sigma(a)-phi(a)+tau(a)-a=n then print(n);
fi; od; end: P(10^6);
  • Mathematica
    f[n_] := If[n > 0, DivisorSigma[1, n] - EulerPhi[n] + DivisorSigma[0, n] - n, 0]; s={}; Do[ If[f[f[n]] == n, AppendTo[s, n]], {n, 1, 60000}]; s (* Amiram Eldar, Mar 31 2019 *)

Extensions

a(11)-a(29) from Amiram Eldar, Mar 31 2019
a(30) from Giovanni Resta, Apr 04 2019

A238228 Numbers n such that if x=sigma(n)+phi(n)-tau(n)-n then n=sigma(x)+phi(x)-tau(x)-x.

Original entry on oeis.org

12, 14, 27, 125, 127
Offset: 1

Views

Author

Paolo P. Lava, Feb 20 2014

Keywords

Comments

a(6) > 2*10^11, if it exists. - Giovanni Resta, Apr 10 2019

Examples

			Fixed points: 27,...
sigma(12) = 28, phi(12) = 4, tau(12) = 6 and 28 + 4 - 6 - 12 = 14.
sigma(14) = 24, phi(14) = 6, tau(14) = 4 and 24 + 6 - 4 - 14 = 12.

Crossrefs

Cf. A000005, A000010, A000203, A238225-A238227, A238229, A238230.

Programs

  • Maple
    with(numtheory); P:=proc(q)local a,n;
for n from 1 to q do a:=sigma(n)+phi(n)-tau(n)-n;
if sigma(a)+phi(a)-tau(a)-a=n then print(n);
fi; od; end: P(10^6);

A238231 Numbers n such that if x=phi(n)*tau(n)-sigma(n)-n then n=phi(x)*tau(x)-sigma(x)-x.

Original entry on oeis.org

376, 594, 846, 1178, 1222, 46498, 65198
Offset: 1

Views

Author

Paolo P. Lava, Feb 20 2014

Keywords

Comments

a(8) > 2*10^9. - Giovanni Resta, Mar 26 2014

Examples

			Fixed points: 376,...
phi(594) = 180, tau(594) = 16, sigma(594) = 1440 and 180*16 - 1440 - 594 = 846.
phi(846) = 276, tau(846) = 12, sigma(846) = 1872 and 276*12 - 1872 - 846 = 594.

Crossrefs

Cf. A000005, A000010, A000203, A238225-A238230.

Programs

  • Maple
    with(numtheory); P:=proc(q)local a,n;
for n from 1 to q do a:=phi(n)*tau(n)-sigma(n)-n;
if a>0 and phi(a)*tau(a)-sigma(a)-a=n then print(n);
fi; od; end: P(10^6);
Showing 1-6 of 6 results.

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