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 1-10 of 22 results. Next

A095955 Function f(x) = phi(sigma(x)) is iterated with initial value n; a(n) is the length of the cycle into which the trajectory merges.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 2, 1, 1, 2, 2, 1, 2, 1, 1, 3, 2, 3, 1, 1, 3, 1, 1, 3, 3, 1, 3, 3, 1, 3, 3, 2, 3, 3, 3, 2, 3, 3, 3, 3, 1, 2, 1, 3, 3, 3, 3, 2, 2, 2, 3, 2, 3, 2, 3, 2, 2, 3, 3, 2, 3, 2, 2, 2, 3, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2
Offset: 1

Views

Author

Labos Elemer, Jul 13 2004

Keywords

Comments

Diagnosis of true cycle of length m: a(j-m) = a(j), but a(j-d) = a(j) cases are excluded for d dividing m.
Length 5 is rare. Example: a(6634509269055173050761216000)=5 and the 5-cycle is {6634509269055173050761216000, 7521613519844726223667200000, 7946886558074859593662464000, 7794495412499746337587200000, 7970172471593905204651622400, 6634509269055173050761216000}. The initial values 2^79 = 604462909807314587353088 and 2^83 = 9671406556917033397649408 after more than 250 transient terms reach this cycle.
a(i) is in {1,2,3,4,6,9,11,12,15,18} for 1 <= i < 254731536. The number 254731536 is the smallest of many integers that are not known to reach a cycle (see the file for a list). - Jud McCranie, Jun 05 2024

Examples

			Occurrences of cycle lengths if n <= 1000: {C1=110, C2=781, C3=36, C4=67, C5=0, C6=6, C7=0, ...}.

Links

Crossrefs

Cf. A000010, A000203, A095952, A096887, A095953, A096526, A095954, A096888, A096889, A096890, A095956.

Programs

  • Mathematica
    g[n_] := EulerPhi[ DivisorSigma[1, n]]; f[n_] := f[n] = Block[{lst = NestWhileList[g, n, UnsameQ, All ]}, -Subtract @@ Flatten[ Position[lst, lst[[ -1]]]]]; Table[ f[n], {n, 105}] (* Robert G. Wilson v, Jul 14 2004 *)
  • PARI
    f(x)=eulerphi(sigma(x))
a(n)=my(t=f(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 22 2013

A095952 Initial values for f(x)=phi(sigma(x)) such that iteration of f ends in cycle of length=1.

Original entry on oeis.org

1, 2, 3, 5, 8, 9, 12, 14, 15, 19, 20, 22, 23, 26, 29, 41, 43, 128, 156, 168, 186, 189, 200, 201, 217, 231, 237, 240, 248, 254, 260, 266, 271, 285, 291, 297, 303, 304, 313, 314, 329, 332, 335, 337, 341, 346, 350, 366, 368, 383, 387, 395, 413, 427, 430, 436, 437
Offset: 1

Views

Author

Labos Elemer, Jul 13 2004

Keywords

Examples

			n=100: trajectory={980, 648, 880, 720, 720}

Links

Crossrefs

Cf. A000010, A000203, A096887, A095953, A096526, A095954, A096888, A096889, A096890, A095955, A095956.

Programs

  • Mathematica
    g[n_] := EulerPhi[ DivisorSigma[1, n]]; f[n_] := f[n] = Block[{lst = NestWhileList[g, n, UnsameQ, All ]}, -Subtract @@ Flatten[ Position[lst, lst[[ -1]]]]]; Select[ Range[1540], f[ # ] == 1 &] (* Robert G. Wilson v, Jul 14 2004 *)
  • PARI
    is(n)=my(t=f(n),h=f(t));while(t!=h,t=f(t);h=f(f(h)));t==f(t) \\ Charles R Greathouse IV, Nov 22 2013

A095953 Initial values for f(x) = phi(sigma(x)) such that iteration of f ends in a cycle of length 3.

Original entry on oeis.org

16, 18, 21, 24, 25, 27, 28, 30, 31, 33, 34, 35, 37, 38, 39, 40, 44, 45, 46, 47, 51, 53, 55, 58, 59, 61, 65, 71, 83, 86, 89, 109, 131, 137, 149, 900, 1116, 1152, 1156, 1200, 1236, 1260, 1300, 1320, 1380, 1386, 1410, 1428, 1458, 1488, 1500, 1518, 1524, 1533, 1536
Offset: 1

Views

Author

Labos Elemer, Jul 13 2004

Keywords

Examples

			n=900: trajectory={900, 2160, 1920, [1536, 1200, 1860], 1536, ...}.

Links

Crossrefs

Cf. A000010, A000203, A095952, A096887, A096526, A095954, A096888, A096889, A096890, A095955, A095956.

Programs

  • Mathematica
    g[n_] := EulerPhi[ DivisorSigma[1, n]]; f[n_] := f[n] = Block[{lst = NestWhileList[g, n, UnsameQ, All ]}, -Subtract @@ Flatten[ Position[lst, lst[[ -1]]]]]; Select[ Range[1540], f[ # ] == 3 &] (* Robert G. Wilson v, Jul 14 2004 *)
fcl3Q[n_]:=Length[FindTransientRepeat[NestList[EulerPhi[DivisorSigma[1,#]]&,n,100],3][[2]]]==3; Select[Range[1600],fcl3Q] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Nov 21 2016 *)
  • PARI
    f(x)=eulerphi(sigma(x))
is(n)=my(t=f(n), h=f(t)); while(t!=h, t=f(t); h=f(f(h))); h=f(h); t!=h && t==f(f(h)) \\ Charles R Greathouse IV, Nov 22 2013

Extensions

More terms from Robert G. Wilson v, Jul 14 2004

A096887 Initial values for f(x)=phi(sigma(x)) such that iteration of f ends in cycle of length=2.

Original entry on oeis.org

4, 6, 7, 10, 11, 13, 17, 32, 36, 42, 48, 49, 50, 52, 54, 56, 57, 60, 62, 63, 64, 66, 67, 68, 69, 70, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 84, 85, 87, 88, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 110, 111, 112, 113, 114
Offset: 1

Views

Author

Labos Elemer and Robert G. Wilson v, Jul 14 2004

Keywords

Links

Crossrefs

Cf. A000010, A000203, A095952, A095953, A096526, A095954, A096888, A096889, A096890, A095955, A095956.

Programs

  • Mathematica
    g[n_] := EulerPhi[ DivisorSigma[1, n]]; f[n_] := f[n] = Block[{lst = NestWhileList[g, n, UnsameQ, All ]}, -Subtract @@ Flatten[ Position[lst, lst[[ -1]]]]]; Select[ Range[115], f[ # ] == 2 &]
  • PARI
    f(x)=eulerphi(sigma(x))
is(n)=my(t=f(n), h=f(t)); while(t!=h, t=f(t); h=f(f(h))); h=f(h); t!=h && t==f(h) \\ Charles R Greathouse IV, Nov 22 2013

A096890 Initial values for f(x)=phi(sigma(x)) such that iteration of f ends in a cycle of length 18.

Original entry on oeis.org

401408, 414050, 436032, 455625, 462400, 466608, 476100, 486300, 486900, 512337, 522242, 526974, 543600, 544644, 544944, 546192, 546861, 554304, 559504, 571536, 572313, 575028, 577200, 579856, 583200, 585528, 599694, 604300, 609429, 611618
Offset: 1

Views

Author

Labos Elemer and Robert G. Wilson v, Jul 14 2004

Keywords

Links

Crossrefs

Cf. A000010, A000203, A095952, A096887, A095953, A096526, A095954, A096888, A096889, A095955, A095956.

Programs

  • Mathematica
    g[n_] := EulerPhi[ DivisorSigma[1, n]]; f[n_] := f[n] = Block[{lst = NestWhileList[g, n, UnsameQ, All ]}, -Subtract @@ Flatten[ Position[lst, lst[[ -1]]]]]; Select[ Range[624900], f[ # ] == 18 &]
  • PARI
    f(x)=eulerphi(sigma(x))
is(n)=my(t=f(n), h=f(t)); while(t!=h, t=f(t); h=f(f(h))); for(i=1,17,h=f(h); if(t==h,return(0))); t==f(h) \\ Charles R Greathouse IV, Nov 25 2013

A095954 Initial values for f(x)=phi(sigma(x)) such that iteration of f ends in a cycle of length 6.

Original entry on oeis.org

784, 800, 882, 912, 960, 972, 1008, 1024, 1050, 1072, 1080, 1089, 1104, 1168, 1204, 1216, 1225, 1232, 1248, 1250, 1264, 1281, 1290, 1296, 1302, 1308, 1332, 1350, 1352, 1360, 1368, 1371, 1372, 1392, 1400, 1407, 1416, 1425, 1440, 1456, 1461, 1464, 1467
Offset: 1

Views

Author

Labos Elemer, Jul 13 2004

Keywords

Examples

			n=882:trajectory={882, 1296, 3300, [2880, 3024, 3840, 3456, 2560, 1800], 2880, ..}

Links

Crossrefs

Cf. A000010, A000203, A095952, A096887, A095953, A096526, A096888, A096889, A096890, A095955, A095956.

Programs

  • Mathematica
    g[n_] := EulerPhi[ DivisorSigma[1, n]]; f[n_] := f[n] = Block[{lst = NestWhileList[g, n, UnsameQ, All ]}, -Subtract @@ Flatten[ Position[lst, lst[[ -1]]]]]; Select[ Range[1500], f[ # ] == 6 &] (* Robert G. Wilson v, Jul 14 2004 *)
  • PARI
    f(x)=eulerphi(sigma(x))
is(n)=my(t=f(n), h=f(t)); while(t!=h, t=f(t); h=f(f(h))); h=f(h); t!=h && t!=(h=f(h)) && t!=(h=f(h)) && t==f(f(f(h))) \\ Charles R Greathouse IV, Nov 23 2013

A096888 Initial values for f(x)=phi(sigma(x)) such that iteration of f ends in cycle of length=11.

Original entry on oeis.org

76050, 88452, 88896, 95922, 104364, 104988, 106032, 108252, 110450, 110928, 112896, 113052, 113412, 113868, 115572, 119892, 119916, 121680, 122220, 122832, 122916, 123060, 123312, 129600, 129840, 129984, 130260, 130560, 131280, 133860
Offset: 1

Views

Author

Labos Elemer and Robert G. Wilson v, Jul 14 2004

Keywords

Links

Crossrefs

Cf. A000010, A000203, A095952, A096887, A095953, A096526, A095954, A096889, A096890, A095955, A095956.

Programs

  • Mathematica
    g[n_] := EulerPhi[ DivisorSigma[1, n]]; f[n_] := f[n] = Block[{lst = NestWhileList[g, n, UnsameQ, All ]}, -Subtract @@ Flatten[ Position[lst, lst[[ -1]]]]]; Select[ Range[134000], f[ # ] == 11 &]
  • PARI
    f(x,k=1)=for(i=1,k,x=eulerphi(sigma(x))); x
is(n)=my(t=f(n), h=f(t)); while(t!=h, t=f(t); h=f(f(h))); h=f(h); t!=h && t==f(h,10) \\ Charles R Greathouse IV, Nov 25 2013

A096889 Initial values for f(x)=phi(sigma(x)) such that iteration of f ends in cycle of length=15.

Original entry on oeis.org

467856, 480200, 490000, 499968, 514416, 523344, 524352, 531441, 548400, 549444, 550512, 553728, 556752, 560532, 562500, 562800, 570000, 570276, 570576, 573744, 576240, 579024, 579900, 581700, 584112, 584836, 586756, 590268, 595008, 599076
Offset: 1

Views

Author

Labos Elemer and Robert G. Wilson v, Jul 14 2004

Keywords

Links

Crossrefs

Cf. A000010, A000203, A095952, A096887, A095953, A096526, A095954, A096888, A096890, A095955, A095956.

Programs

  • Mathematica
    g[n_] := EulerPhi[ DivisorSigma[1, n]]; f[n_] := f[n] = Block[{lst = NestWhileList[g, n, UnsameQ, All ]}, -Subtract @@ Flatten[ Position[lst, lst[[ -1]]]]]; Select[ Range[599800], f[ # ] == 15 &]
  • PARI
    f(x,k=1)=for(i=1,k,x=eulerphi(sigma(x))); x
is(n)=my(t=f(n), h=f(t)); while(t!=h, t=f(t); h=f(f(h))); h=f(h); t!=h && t!=(h=f(h,2)) && t!=(h=f(h,2)) && t==f(h,10) \\ Charles R Greathouse IV, Nov 25 2013

A096526 Initial values for f(x)=phi(sigma(x)) such that iteration of f ends in a cycle of length 4.

Original entry on oeis.org

324, 400, 484, 490, 530, 544, 576, 630, 660, 672, 684, 690, 714, 722, 750, 756, 768, 770, 772, 777, 780, 792, 804, 810, 819, 828, 832, 833, 840, 841, 846, 852, 858, 864, 868, 870, 872, 876, 888, 892, 901, 906, 910, 918, 920, 924, 930, 936, 940, 948, 952
Offset: 1

Views

Author

Labos Elemer, Jul 13 2004

Keywords

Examples

			n=324: trajectory=324, 660, [576, 1512, 1280, 864], 576, ...}

Links

Crossrefs

Cf. A000010, A000203, A095952, A096887, A095953, A095954, A096888, A096889, A096890, A095955, A095956.

Programs

  • Mathematica
    g[n_] := EulerPhi[ DivisorSigma[1, n]]; f[n_] := f[n] = Block[{lst = NestWhileList[g, n, UnsameQ, All ]}, -Subtract @@ Flatten[ Position[lst, lst[[ -1]]]]]; Select[ Range[1000], f[ # ] == 4 &] (* Robert G. Wilson v, Jul 14 2004 *)
  • PARI
    f(x)=eulerphi(sigma(x))
is(n)=my(t=f(n), h=f(t)); while(t!=h, t=f(t); h=f(f(h))); h=f(h); t!=h && t!=(h=f(h)) && t==f(f(h)) \\ Charles R Greathouse IV, Nov 22 2013

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
Showing 1-10 of 22 results. Next

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