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.

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A095397 Modified juggler map: see A095396. Largest value in trajectory of started n under the juggler map of A095396.

Original entry on oeis.org

1, 2, 36, 4, 36, 36, 36, 8, 140, 10, 36, 36, 46, 36, 58, 36, 70, 36, 82, 36, 96, 36, 110, 24, 52214, 26, 140, 140, 156, 140, 172, 32, 2598, 34, 2978, 36, 86818724, 38, 233046, 40, 262, 42, 4710, 44, 5222, 46, 322, 48, 6352, 50, 364, 52, 7554, 54, 8210, 56, 430, 58
Offset: 1

Views

Author

Labos Elemer, Jun 18 2004

Keywords

Comments

Parallel to A094716.

Examples

			n=37: the trajectory is {37, 225, 3375, 196069, 86818724, 196068, 3374, 224, 36, 10, 4, 2, 1}, the peak is a[37]=86818724

Crossrefs

Cf. A007320, A094683, A094716, A095396, A097398.

Programs

  • Mathematica
    d[x_]:=d[x]=(1-Mod[x, 2])*Floor[N[x^(2/3), 50]] +Mod[x, 2]*Floor[N[x^(3/2), 50]];d[1]=1; fd[x_]:=Delete[FixedPointList[d, x], -1] Table[Max[fd[w]], {w, 1, m}]

A095398 Number of steps required to reach 1 for iterated modified juggler map of A095396.

Original entry on oeis.org

0, 1, 7, 2, 6, 8, 10, 3, 7, 3, 5, 7, 9, 7, 9, 9, 11, 9, 11, 11, 13, 11, 13, 4, 10, 4, 6, 8, 10, 8, 10, 4, 8, 4, 8, 4, 12, 6, 12, 6, 8, 8, 12, 8, 12, 8, 10, 10, 14, 10, 12, 10, 14, 8, 12, 8, 10, 8, 14, 10, 12, 10, 12, 10, 12, 10, 12, 10, 14, 10, 12, 12, 16, 12, 18, 12, 18, 10, 12, 10, 16
Offset: 1

Views

Author

Labos Elemer, Jun 18 2004

Keywords

Comments

Parallel to A007320.

Examples

			n=37: the trajectory is {37, 225, 3375, 196069, 86818724, 196068, 3374, 224, 36, 10, 4, 2, 1}, number of required steps is a[37]=13-1=12.

Crossrefs

Cf. A007320, A094683, A094716, A095396, A097397.

Programs

  • Mathematica
    d[x_]:=d[x]=(1-Mod[x, 2])*Floor[N[x^(2/3), 50]] +Mod[x, 2]*Floor[N[x^(3/2), 50]];d[1]=1; fd[x_]:=Delete[FixedPointList[d, x], -1] Table[Max[fd[w]], {w, 1, m}]
Table[Length[NestWhileList[If[EvenQ[#],Floor[#^(2/3)],Floor[#^(3/2)]]&, n, #!=1&]]-1,{n,90}] (* Harvey P. Dale, Dec 28 2018 *)

A143745 The next largest juggler number.

Original entry on oeis.org

1, 2, 36, 140, 52214, 24906114455136, 202924588924125339424550328
Offset: 1

Views

Author

Harry J. Smith, Oct 08 2008

Keywords

Comments

The juggler sequence: begin with a starting value x and if x is even, x <- floor(sqrt(x)) and if x is odd, x <- floor(sqrt(x^3)) and repeat until x = 1, save the starting value, max x and the number of steps needed to reach it.
I have a b-file for this sequence for n=1,...,19 (all known values), but some a(n) values are much larger than 1000 digits.

Examples

			24906114455136 is in the sequence because starting at 37 the juggler sequences maxes out at 24906114455136, a 14-digit number, after 8 steps. This is the largest juggler number found for starting values less than or equal to 37.

References

  • C. Pickover, Computers and the Imagination, St. Martin's Press, NY, 1991, p. 233.

Links

Crossrefs

Cf. A007320, A094670, A094679, A094683, A094684, A095908.

A094778 "Dropping time" in juggler sequence problem starting at 2n+1 (number of steps to reach a lower number than starting value).

Original entry on oeis.org

0, 5, 4, 2, 5, 2, 2, 2, 2, 2, 2, 2, 5, 2, 2, 2, 4, 4, 15, 5, 2, 4, 4, 2, 5, 2, 4, 4, 2, 5, 2, 2, 2, 2, 7, 2, 4, 5, 10, 2, 7, 2, 4, 7, 8, 2, 2, 5, 5, 8, 5, 13, 8, 2, 7, 8, 13, 10, 4, 2, 4, 2, 4, 4, 8, 4, 4, 2, 5, 2, 2, 2, 2, 2, 2, 4, 2, 5, 5, 2, 2, 24, 10, 2, 4, 2, 26, 5, 2, 2, 4, 10, 2, 5, 2, 4, 70, 4, 5, 5, 5
Offset: 0

Views

Author

Jason Earls, Jun 10 2004

Keywords

Comments

If the starting value is even then of course the next step in the trajectory is smaller.

Examples

			9 -> 27 -> 140 -> 11 -> 36 -> 6, taking 5 steps, so a(4) = 5.

Crossrefs

Cf. A007320, A094683.

A095399 Modified juggler modified further: a[n]=(1-Mod[n,2])*Floor[n^(3/4)]+Mod[n,2]*Floor[n^(4/3)]; original exponents {1/2,3/2} are replaced with {3/4,4/3}.

Original entry on oeis.org

1, 1, 4, 2, 8, 3, 13, 4, 18, 5, 24, 6, 30, 7, 36, 8, 43, 8, 50, 9, 57, 10, 65, 10, 73, 11, 81, 12, 89, 12, 97, 13, 105, 14, 114, 14, 123, 15, 132, 15, 141, 16, 150, 17, 160, 17, 169, 18, 179, 18, 189, 19, 199, 19, 209, 20, 219, 21, 229, 21, 240, 22, 250, 22, 261, 23, 272, 23
Offset: 1

Views

Author

Labos Elemer, Jun 18 2004

Keywords

Crossrefs

Cf. A007320, A094683, A094716, A094396-A094401.

Programs

  • Mathematica
    e[x_]:=e[x]=(1-Mod[x, 2])*Floor[N[x^(3/4), 50]] +Mod[x, 2]*Floor[N[x^(4/3), 50]];e[1]=1; Table[e[w], {w, 1, 150}]

A095400 Largest value in trajectory when the following modified juggler map is iterated: a[n]=(1-Mod[n, 2])*Floor[n^(3/4)]+Mod[n, 2]*Floor[n^(4/3)]; original exponents {1/2, 3/2} are replaced with {3/4, 4/3}.

Original entry on oeis.org

1, 2, 4, 4, 8, 6, 30, 8, 18, 10, 24, 12, 30, 30, 36, 16, 150, 18, 50, 20, 1320, 22, 43366048, 24, 26092, 26, 350, 28, 41678, 30, 234421146, 32, 2438232, 34, 114, 36, 5184, 38, 132, 40, 124026, 42, 150, 150, 160, 150, 934, 48, 1008, 50, 1084, 52, 12202, 54, 1240, 56
Offset: 1

Views

Author

Labos Elemer, Jun 18 2004

Keywords

Examples

			n=101: the trajectory is {101, 470, 100, 31, 97, 445, 3397, 51065, 1894513, 234421146, 1894512, 51064, 3396, 444, 96, 30, 12, 6, 3, 4, 2, 1}, peak=a[101]=234421146.

Crossrefs

Cf. A007320, A094683, A094716, A094396-A094401.

Programs

  • Mathematica
    e[x_]:=e[x]=(1-Mod[x, 2])*Floor[N[x^(3/4), 50]] +Mod[x, 2]*Floor[N[x^(4/3), 50]];e[1]=1; fe[x_]:=Delete[FixedPointList[e, x], -1]; Table[Max[fe[w]], {w, 1, 150}]

A095401 number of steps required to reach 1 if the following modified juggler map is iterated: a[n]=(1-Mod[n, 2])*Floor[n^(3/4)]+Mod[n, 2]*Floor[n^(4/3)]; original exponents {1/2, 3/2} are replaced with {3/4, 4/3}.

Original entry on oeis.org

0, 1, 3, 2, 4, 4, 8, 3, 5, 5, 7, 5, 7, 9, 11, 4, 8, 4, 6, 6, 12, 6, 18, 6, 14, 8, 12, 6, 14, 6, 18, 8, 18, 10, 12, 10, 16, 12, 14, 12, 20, 5, 7, 9, 11, 9, 13, 5, 9, 5, 9, 7, 13, 7, 11, 7, 11, 13, 19, 13, 15, 7, 9, 7, 17, 19, 21, 19, 23, 7, 11, 7, 13, 15, 17, 15, 19, 9, 11, 9, 11, 13, 15, 13, 19
Offset: 1

Views

Author

Labos Elemer, Jun 18 2004

Keywords

Examples

			n=101: the trajectory is {101, 470, 100, 31, 97, 445, 3397, 51065, 1894513, 234421146, 1894512, 51064, 3396, 444, 96, 30, 12, 6, 3, 4, 2, 1}, number of required steps is a[101]=22-1=21.

Crossrefs

Cf. A007320, A094683, A094716, A094396-A094400.

Programs

  • Mathematica
    e[x_]:=e[x]=(1-Mod[x, 2])*Floor[N[x^(3/4), 50]] +Mod[x, 2]*Floor[N[x^(4/3), 50]];e[1]=1; fe[x_]:=Delete[FixedPointList[e, x], -1]; Table[ -1+Length[fe[w]], {w, 1, 150}]

A143742 Starting values that produce a larger juggler number than smaller starting values.

Original entry on oeis.org

1, 2, 3, 9, 25, 37, 113, 173, 193, 2183, 11229, 15065, 15845, 30817, 48443, 275485, 1267909, 2264915, 5812827, 7110201
Offset: 1

Views

Author

Harry J. Smith, Oct 06 2008

Keywords

Comments

The juggler sequence: begin with a starting value x and if x is even, x -> [sqrt(x)] and if x is odd, x -> [sqrt(x^3)] and repeat until x = 1, save the starting value, max x and the number of steps needed to reach it.

Examples

			37 is in the sequence because starting at 37 the juggler sequences maxes out at 24906114455136, a 14-digit number, after 8 steps. This is the largest juggler number found for starting values less than or equal to 37.

References

  • C. Pickover, Computers and the Imagination, St. Martin's Press, NY, 1991, p. 233.

Links

Crossrefs

Cf. A007320, A094670, A094679, A094683, A094684, A095908.

Programs

  • Mathematica
    DeleteDuplicates[Table[{n,Max[NestWhileList[If[EvenQ[#],Floor[Sqrt[#]],Floor[Sqrt[#^3]]]&,n,#!=1&]]},{n,50000}],GreaterEqual [#1[[2]],#2[[2]]]&][[;;,1]] (* The program generates the first 15 terms of the sequence. *) (* Harvey P. Dale, Dec 21 2024 *)

Extensions

Comment clarified by Harvey P. Dale, Dec 21 2024

A143743 The number of digits in the next largest juggler number.

Original entry on oeis.org

1, 1, 2, 3, 5, 14, 27, 82, 271, 5929, 8201, 11723, 23889, 45391, 972463, 1909410, 1952329, 2855584, 7996276
Offset: 1

Views

Author

Harry J. Smith, Oct 08 2008

Keywords

Comments

The juggler sequence: begin with a starting value x and if x is even, x <- [sqrt(x)] and if x is odd, x <- [sqrt(x^3)] and repeat until x = 1, save the starting value, max x and the number of steps needed to reach it.

Examples

			14 is in the sequence because starting at 37 the juggler sequence maxes out at 24906114455136, a 14-digit number, after 8 steps. This is the largest juggler number found for starting values less than or equal to 37.

References

  • C. Pickover, Computers and the Imagination, St. Martin's Press, NY, 1991, p. 233.

Links

Crossrefs

Cf. A007320, A094670, A094679, A094683, A094684, A095908.

A143744 The number of steps needed to generate the next largest juggler number.

Original entry on oeis.org

0, 0, 3, 2, 3, 8, 9, 17, 47, 32, 54, 25, 43, 39, 60, 148, 99, 89, 67
Offset: 1

Views

Author

Harry J. Smith, Oct 08 2008

Keywords

Comments

The juggler sequence: begin with a starting value x and if x is even, x <- [sqrt(x)] and if x is odd, x <- [sqrt(x^3)] and repeat until x = 1, save the starting value, max x and the number of steps needed to reach it.

Examples

			8 is in the sequence because starting at 37 the juggler sequences maxes out at 24906114455136, a 14-digit number, after 8 steps. This is the largest juggler number found for starting values less than or equal to 37.

References

  • C. Pickover, Computers and the Imagination, St. Martin's Press, NY, 1991, p. 233.

Links

Crossrefs

Cf. A007320, A094670, A094679, A094683, A094684, A095908.
Previous Showing 11-20 of 21 results. Next

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