A094679 -id:A094679 - cp's OEIS frontend

A007320 Number of steps needed for juggler sequence (A094683) started at n to reach 1.

0, 1, 6, 2, 5, 2, 4, 2, 7, 7, 4, 7, 4, 7, 6, 3, 4, 3, 9, 3, 9, 3, 9, 3, 11, 6, 6, 6, 9, 6, 6, 6, 8, 6, 8, 3, 17, 3, 14, 3, 5, 3, 6, 3, 6, 3, 6, 3, 11, 5, 11, 5, 11, 5, 11, 5, 5, 5, 11, 5, 11, 5, 5, 3, 5, 3, 11, 3, 14, 3, 5, 3, 8, 3, 8, 3, 19, 3, 8, 3, 10, 8, 8, 8, 11, 8, 10, 8, 11, 8, 11, 8, 11, 8, 8, 8, 11

Offset: 1

N. J. A. Sloane, Robert G. Wilson v, Mira Bernstein

nonn

It is not known if every starting value eventually reaches 1.

			The trajectory of 1 is 3, 5, 11, 36, 6, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ... so a(3) = 6.

C. Pickover, Computers and the Imagination, St. Martin's Press, NY, 1991, p. 232.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Giovanni Resta, Table of n, a(n) for n = 1..10000
H. J. Smith, Juggler Sequence
Eric Weisstein's World of Mathematics, Juggler Sequence
Wikipedia, Juggler sequence
R. G. Wilson, V, Letter to N. J. A. Sloane, Sep. 1992

Cf. A007321, A094683, A094698, A094679, A093685, A094716.

A007320 := proc(n)
    local a,ntrack;
    a := 0 ;
    ntrack := n ;
    while ntrack > 1 do
        ntrack := A094683(ntrack) ;
        a := a+1 ;
    end do:
    return a;
end proc: # R. J. Mathar, Apr 19 2013

js[n_] := If[ EvenQ[n], Floor[ Sqrt[n]], Floor[ Sqrt[n^3]]]; f[n_] := Length[ NestWhileList[js, n, # != 1 &]] - 1; Table[ f[n], {n, 99}] (* Robert G. Wilson v, Jun 10 2004 *)

Corrected and extended by Jason Earls, Jun 09 2004

A094670 Smallest number which requires n iterations to reach 1 in the juggler sequence problem.

Original entry on oeis.org

1, 2, 4, 16, 7, 5, 3, 9, 33, 19, 81, 25, 353, 183, 39, 201, 103, 37, 205, 77, 681, 263, 3817, 429, 175, 1673, 539, 165, 671, 321, 5875, 477, 173, 2243, 265, 29017, 1011, 677, 9361, 659, 241, 3389, 1123, 163, 2057, 625, 15271, 4481

Offset: 0

Jason Earls, Jun 09 2004

nonn

A juggler sequence is defined as follows: given a positive integer x, repeat: if x is even then x <- [x^(1/2)] else x <- [x^(3/2)] until x=1. The brackets indicate the floor function.

a(104) is unknown ( > 10000000). - Robert G. Wilson v, Jun 11 2014

Robert G. Wilson v, Table of n, a(n) for n = 0..103
Eric Weisstein's World of Mathematics, Juggler Sequence
Robert G. Wilson v, List of n & a(n) for n = 0..450 (with 0's for unknown entries)

Cf. A007320, A094679, A094698.

js[n_] := If[ EvenQ[ n], Floor[ Sqrt[n]], Floor[ Sqrt[n^3]]]; f[n_] := Length[ NestWhileList[js, n, # != 1 &]] - 1; a = Table[0, {50}]; Do[ b = f[n]; If[b < 51 && a[[b]] == 0, a[[b]] = n; Print[n, " = ", b]], {n, 10^5}] (* Robert G. Wilson v *)

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

A094698 Number of steps where the Juggler sequence reaches a new record.

Original entry on oeis.org

0, 1, 6, 7, 9, 11, 17, 19, 43, 73, 75, 80, 88, 96, 107, 131, 166, 193, 201, 258, 263, 268, 271, 298, 335, 340, 443, 479, 484, 527

Offset: 1

N. J. A. Sloane, Jun 09 2004

Records in A007320.

The Juggler sequence: begin with x; if x is even, floor(sqrt(x)) -> x; if x is odd, floor(sqrt(x^3)) -> x; repeat until x = 1, count the iterations.

Eric Weisstein's World of Mathematics, Juggler Sequence

Cf. A007320, A094670, A094679.

$MaxPrecision = 250000000; js[n_] := If[ EvenQ[ n], Floor[ Sqrt[n]], Floor[ Sqrt[n^3]]]; f[n_] := Block[{c = 1, k = n}, While[k = js[k]; k != 1, c++ ]; c]; a = {0}; Do[ b = f[n]; If[b > a[[ -1]], AppendTo[a, b]; Print[n]], {n, 3053595}] (* Robert G. Wilson v, Jun 14 2004 *)

More terms from Robert G. Wilson v, Jun 14 2004
a(25) = 335 from Eric W. Weisstein, Jan 25 2006
Edited by N. J. A. Sloane, Sep 16 2008 at the suggestion of Tim Nikkel
a(26)-a(29) from Giovanni Resta, Apr 08 2017
a(30) from Ethan Slota, Apr 15 2025

A143745 The next largest juggler number.

Original entry on oeis.org

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

Offset: 1

Harry J. Smith, Oct 08 2008

nonn

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.

			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.

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

Harry J. Smith, Table of n, a(n) for n = 1..9
H. J. Smith, Juggler Sequence
Eric Weisstein's World of Mathematics, Juggler Sequence

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

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

Harry J. Smith, Oct 06 2008

nonn

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.

			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.

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

Eric Weisstein's World of Mathematics, Juggler Sequence
H. J. Smith, Juggler Sequence

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

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 *)

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

Harry J. Smith, Oct 08 2008

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.

			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.

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

H. J. Smith, Juggler Sequence
Eric Weisstein's World of Mathematics, Juggler Sequence

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

Harry J. Smith, Oct 08 2008

nonn

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.

			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.

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

Eric Weisstein's World of Mathematics, Juggler Sequence
H. J. Smith, Juggler Sequence

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

cp's OEIS Frontend

A007320 Number of steps needed for juggler sequence (A094683) started at n to reach 1.

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A094670 Smallest number which requires n iterations to reach 1 in the juggler sequence problem.

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A094698 Number of steps where the Juggler sequence reaches a new record.

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A143745 The next largest juggler number.

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

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Programs

Mathematica

Extensions

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

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Author

Keywords

Comments

Examples

References

Links

Crossrefs

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

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs