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.

A096852 a(n) is the length of terminal cycle of the trajectory of f(x)=phi(sigma(x)) if started at 2^n.

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 2, 1, 2, 2, 6, 2, 1, 6, 2, 1, 2, 3, 11, 11, 2, 2, 15, 15, 18, 18, 18, 18, 12, 12, 12, 1
Offset: 0

Views

Author

Labos Elemer, Jul 16 2004

Keywords

Examples

			n=18: start = 262144 and the corresponding 11-cycle is 262144, 524286, [368640, 381024, 326592, 550368, 435456, 580608, 851840, 552960, 524160, 442368, 432000], 368640, ...

Crossrefs

Cf. A095955, A095956, A096849, A096850.

Programs

  • Mathematica
    g[n_] := EulerPhi[ DivisorSigma[1, n]]; f[n_] := Block[{lst = NestWhileList[g, n, UnsameQ, All]}, -Subtract @@ Flatten[ Position[lst, lst[[ -1]]]]]; Table[ f[2^n], {n, 0, 20}]
  • PARI
    f(x)=eulerphi(sigma(x))
a(n)=my(t=f(2^n), h=f(t), s); while(t!=h, t=f(t); h=f(f(h))); t=f(t); h=f(t); s=1; while(t!=h, s++; t=f(t); h=f(f(h))); s \\ Charles R Greathouse IV, Nov 27 2013

Formula

a(n) = A095955(2^n). - Charles R Greathouse IV, Nov 27 2013

Extensions

Edited, corrected and extended by Robert G. Wilson v, Jul 17 2004

A096850 Consider iteration of the function f(x) = phi(sigma(x)) = A062401(x). Sequence gives numbers n such that the trajectory of n returns to n.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 16, 24, 30, 48, 60, 72, 96, 128, 240, 432, 480, 576, 720, 864, 1200, 1280, 1512, 1536, 1728, 1800, 1860, 2016, 2560, 2880, 3024, 3456, 3840, 6912, 10368, 14080, 15552, 15840, 18144, 27648, 30976, 32768, 34560, 41472, 42240, 48384
Offset: 1

Views

Author

Labos Elemer, Jul 16 2004

Keywords

Examples

			Examples of cycles: {[1], [2], [4, 6], [8], [12], [16, 30, 24], [48, 60], [72, 96], [128]}.
95 => 32 => 36 => 72 => 96 => 72 => ..., therefore 72 and 96 are in the sequence.

Links

Crossrefs

Cf. A062401, A095952-A095956, A096887-A096890, A096849-A096851.

Programs

  • Mathematica
    a = {}; f[n_] := EulerPhi[ DivisorSigma[ 1, n]]; Do[ AppendTo[a, NestWhileList[f, n, UnsameQ, All][[ -1]]]; a = Union[a], {n, 10^6}]; Take[ a, 46] (* Robert G. Wilson v, Jul 21 2004 *)
  • PARI
    f(n)=eulerphi(sigma(n))
is(n)=my(t=f(n),h=f(t));while(t!=h,t=f(t);h=f(f(h));if(t==n, return(1)));t==n \\ Charles R Greathouse IV, Nov 27 2013

Extensions

Edited and extended by Robert G. Wilson v, Jul 21 2004

A096851 Even transient values of f(x)=phi(sigma(x)) iterations.

Original entry on oeis.org

10, 14, 18, 20, 22, 26, 28, 32, 34, 36, 38, 40, 42, 44, 46, 50, 52, 54, 56, 58, 62, 64, 66, 68, 70, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 98, 100, 102, 104, 106, 108, 110, 112, 114, 116, 118, 120, 122, 124, 126
Offset: 1

Views

Author

Labos Elemer, Jul 16 2004

Keywords

Links

Crossrefs

Even terms of A096849.
Cf. A095952-A095956, A096887-A096890, A096849-A096850.

A096997 If the function f(x) = sigma(phi(x)) = A062402(x) is iterated starting from these listed values, then the starting value never returns. These are the transient terms of this iteration; they never occur in terminal cycles.

Original entry on oeis.org

2, 4, 5, 6, 8, 9, 10, 11, 13, 14, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 29, 30, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 80
Offset: 1

Views

Author

Labos Elemer, Jul 19 2004

Keywords

Crossrefs

Cf. A062402, A096849.
Showing 1-4 of 4 results.

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