A007320 -id:A007320 - cp's OEIS frontend

A143745 The next largest juggler number.

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.

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

Jason Earls, Jun 10 2004

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

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

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

Labos Elemer, Jun 18 2004

nonn

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

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

Labos Elemer, Jun 18 2004

nonn

			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.

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

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

Labos Elemer, Jun 18 2004

nonn

			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.

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

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

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.

A284987 Juggler sequence starting at 37.

Original entry on oeis.org

37, 225, 3375, 196069, 86818724, 9317, 899319, 852846071, 24906114455136, 4990602, 2233, 105519, 34276462, 5854, 76, 8, 2, 1

Offset: 0

Matt Westwood, Apr 07 2017

Many sources state that the Juggler sequence on 37 is, for example, "the first tall peak in its graph", and "the high water number of steps are" and so on, without specifically giving the actual sequence. As this specific instance is cited in various places as a specific instance, if not a special case, it seems worthy to document it here explicitly.

David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Revised Edition, Penguin Books, 1997, page 99.

Eric Weisstein's World of Mathematics, Juggler Sequence
Calculated using the Juggler Sequence calculator found at Calculation for it

Cf. A007320, A094670, A094683.

Juggler(n)={my(L=List([n])); while(n<>1, n=sqrtint(n^(2-(-1)^n)); listput(L,n)); Vec(L)}
{ Juggler(37) } \\ Andrew Howroyd, Apr 27 2020

a(0) = 37, a(n) = floor(sqrt(a(n-1))): n even, a(n) = floor((sqrt(a(n-1)))^3): n odd.

cp's OEIS Frontend

A143745 The next largest juggler number.

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

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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}.

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Mathematica

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}.

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Mathematica

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}.

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Mathematica

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

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Mathematica

Extensions

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

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

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A284987 Juggler sequence starting at 37.

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PARI

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}.

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}.

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}.