A007320 -id:A007320 - cp's OEIS frontend

A094683 Juggler sequence: if n mod 2 = 0 then floor(sqrt(n)) else floor(n^(3/2)).

0, 1, 1, 5, 2, 11, 2, 18, 2, 27, 3, 36, 3, 46, 3, 58, 4, 70, 4, 82, 4, 96, 4, 110, 4, 125, 5, 140, 5, 156, 5, 172, 5, 189, 5, 207, 6, 225, 6, 243, 6, 262, 6, 281, 6, 301, 6, 322, 6, 343, 7, 364, 7, 385, 7, 407, 7, 430, 7, 453, 7, 476, 7, 500, 8, 524, 8, 548, 8, 573, 8, 598, 8, 623, 8, 649

Offset: 0

N. J. A. Sloane, Jun 09 2004

nonn

Interspersion of A000093 and A000196. - Michel Marcus, Nov 11 2013

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

Karl V. Boddy, Table of n, a(n) for n = 0..10000
Vikram Prasad and M. A. Prasad, Estimates of the maximum excursion constant and stopping constant of juggler-like sequences, ResearchGate, 2025.
Harry J. Smith, Juggler Sequence
Eric Weisstein's World of Mathematics, Juggler Sequence
Wikipedia, Juggler sequence

Cf. A007320, A007321, A094684, A094685, A094716.

Cf. A000093, A000196.

A094683 :=proc(n) if n mod 2 = 0 then RETURN(floor(sqrt(n))) else RETURN(floor(n^(3/2))); end if; end proc;

Table[If[EvenQ[n], Floor[Sqrt[n]], Floor[n^(3/2)]], {n, 0, 100}] (* Indranil Ghosh, Apr 07 2017 *)

a(n) = if(n%2,sqrtint(n^3), sqrtint(n)); \\ Indranil Ghosh, Apr 08 2017

import math
from sympy import sqrt
def a(n): return int(math.floor(sqrt(n))) if n%2 == 0 else int(math.floor(n**(3/2)))
print([a(n) for n in range(51)]) # Indranil Ghosh, Apr 08 2017

from math import isqrt
def A094683(n): return isqrt(n**3 if n % 2 else n) # Chai Wah Wu, Feb 18 2022

A007321 Number of steps needed for modified juggler sequence (A094685) started at n to reach 1.

Original entry on oeis.org

0, 1, 6, 2, 5, 2, 13, 7, 10, 7, 4, 7, 6, 3, 9, 3, 9, 3, 12, 3, 9, 6, 9, 6, 19, 6, 9, 6, 9, 6, 16, 3, 5, 3, 8, 3, 16, 3, 5, 3, 14, 3, 11, 14, 11, 14, 5, 14, 14, 14, 14, 14, 5, 14, 5, 14, 11, 8, 11, 8, 8, 8, 8, 8, 11, 8, 11, 8, 8, 8, 8, 8, 21, 11, 21, 11, 8, 11, 8, 11, 19, 11, 11, 11, 8, 11, 11, 11, 11

Offset: 1

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

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

C. Pickover, Computers and the Imagination, St. Martin's Press, NY, 1991, p. 233.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Richard J Mathar and Chai Wah Wu, Table of n, a(n) for n = 1..10000 n = 1..836 from Richard J Mathar
H. J. Smith, Juggler Sequence
R. G. Wilson, V, Letter to N. J. A. Sloane, Sep. 1992

Cf. A094685, A007320, A094683.

f:=proc(n) if n mod 2 = 0 then RETURN(round(sqrt(n))) else RETURN(round(n^(3/2))); fi; end; # corrected by R. J. Mathar, Jul 28 2007

mjs[n_] := If[EvenQ[n], Round[Sqrt[n]], Round[Sqrt[n^3]]]; f[n_] := Length[NestWhileList[mjs, n, # != 1 &]] - 1; Table[ f[n], {n, 90}]

More terms from N. J. A. Sloane, 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

A094679 n sets a new record for number of iterations to reach 1 in the juggler sequence problem.

Original entry on oeis.org

1, 2, 3, 9, 19, 25, 37, 77, 163, 193, 1119, 1155, 4065, 4229, 4649, 7847, 13325, 34175, 59739, 78901, 636731, 1122603, 1301535, 2263913, 5947165, 72511173, 78641579, 125121851, 198424189, 4488817391

Offset: 1

Jason Earls, Jun 09 2004

Where records occur in A007320.
The Juggler sequence: begin with x and if x is even, [sqrt(x)] -> x and if x is odd, [sqrt(x^3)] -> x and repeat until x = 1, count the iterations. - Robert G. Wilson v, Jun 14 2004
78901 reaches a maximum of 4064983429...(skip the next 371727 digits)...2140697134 during its trip to 1. - Robert G. Wilson v, Jun 14 2004
I postulate that 2 is the only even number in this sequence. - Harry J. Smith, Aug 15 2008
a(30) > 1.6*10^9. - Giovanni Resta, Apr 08 2017

			78901 takes 258 iterations to reach 1; see A094698 for the others.

Eric Weisstein's World of Mathematics, Juggler Sequence

Cf. A007320, A094670, A094698, A095906, A094698.

$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]], {n, 3053595}]; a (* Robert G. Wilson v *)

More terms from Robert G. Wilson v, Jun 14 2004
a(25) = 5947165 from Eric W. Weisstein, Jan 25 2006
a(26)-a(27) from Robert G. Wilson v, Jun 15 2014
a(28)-a(29) from Giovanni Resta, Apr 08 2017
a(30) from Ethan Slota, Apr 15 2025

A094685 Modified juggler sequence: if n mod 2 = 0 then round(sqrt(n)) else round(n^(3/2)).

Original entry on oeis.org

0, 1, 1, 5, 2, 11, 2, 19, 3, 27, 3, 36, 3, 47, 4, 58, 4, 70, 4, 83, 4, 96, 5, 110, 5, 125, 5, 140, 5, 156, 5, 173, 6, 190, 6, 207, 6, 225, 6, 244, 6, 263, 6, 282, 7, 302, 7, 322, 7, 343, 7, 364, 7, 386, 7, 408, 7, 430, 8, 453, 8, 476, 8, 500, 8, 524, 8, 548, 8, 573, 8, 598, 8, 624, 9, 650

Offset: 0

N. J. A. Sloane, Jun 09 2004

nonn

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

Chai Wah Wu, Table of n, a(n) for n = 0..10000
Vikram Prasad and M. A. Prasad, Estimates of the maximum excursion constant and stopping constant of juggler-like sequences, ResearchGate, 2025.
Harry J. Smith, Juggler Sequence

Cf. A007320, A007321, A094684, A094683, A094693, A094725.

f:=proc(n) if n mod 2 = 0 then RETURN(round(sqrt(n))) else RETURN(round(n^(3/2))); fi; end;

A094685[n_]:=If[Mod[n,2]==0,Round[Sqrt[n]],Round[n^(3/2)]];Array[A094685,76,0] (* James C. McMahon, Apr 15 2025 *)

from gmpy2 import isqrt_rem
def A094685(n):
    i, j = isqrt_rem(n**3 if n % 2 else n)
    return int(i+ int(4*(j-i) >= 1)) # Chai Wah Wu, Aug 17 2016

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

A095396 Modified juggler map: for even numbers: a(n) = floor(n^(2/3)) and for odd n: a(n) = floor(n^(3/2)) = floor(sqrt(n^3)).

Original entry on oeis.org

1, 1, 5, 2, 11, 3, 18, 4, 27, 4, 36, 5, 46, 5, 58, 6, 70, 6, 82, 7, 96, 7, 110, 8, 125, 8, 140, 9, 156, 9, 172, 10, 189, 10, 207, 10, 225, 11, 243, 11, 262, 12, 281, 12, 301, 12, 322, 13, 343, 13, 364, 13, 385, 14, 407, 14, 430, 14, 453, 15, 476, 15, 500, 16, 524, 16, 548, 16

Offset: 1

Labos Elemer, Jun 18 2004

nonn

Parallel to A094683: values for odd arguments are same, for even not necessarily so.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000196, A007320, A048766, A094683, A094716, A095397, A095398.

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; Table[d[w], {w, 1, 150}]
Table[If[EvenQ[n],Floor[n^(2/3)],Floor[n^(3/2)]],{n,70}] (* Harvey P. Dale, Dec 28 2018 *)

(define (A095396 n) (if (even? n) (A048766 (* n n)) (A000196 (* n n n)))) ;; Antti Karttunen, May 28 2017

For even n: a(n) = A048766(n^2), for odd n: a(n) = A000196(n^3). - Antti Karttunen, May 28 2017

Name simplified by Antti Karttunen, May 28 2017

A094819 Number of steps needed for juggler sequence (A094683) started at 10^n to reach 1.

Original entry on oeis.org

0, 7, 8, 7, 9, 6, 8, 7, 10, 7, 7, 20, 9, 25, 8, 11, 11, 13, 8, 10, 8, 16, 21, 24, 10, 18, 26, 15, 9, 45, 12, 31, 12, 20, 14, 22, 9, 20, 11, 25, 9, 33, 17, 33, 22, 30, 25, 11, 11, 30, 19, 13, 27, 21, 16, 10, 10, 16, 46, 70, 13, 13, 32, 21, 13, 21, 21, 48, 15, 29, 23, 29, 10, 18, 21

Offset: 0

Jason Earls, Jun 12 2004

nonn

Cf. A007320, A011557, A094683.

fj[n_]:=If[EvenQ[n], Floor[Sqrt[n]], Floor[n^(3/2)]];a094819[n_]:=Length[ NestWhileList[fj,10^n, # != 1 &]] - 1;Array[a094819,75,0] (* James C. McMahon, Apr 18 2025 *)

a(n) = A007320(A011557(n)). - Michel Marcus, Apr 19 2025

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

Labos Elemer, Jun 18 2004

nonn

Parallel to A094716.

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

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

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

Labos Elemer, Jun 18 2004

nonn

Parallel to A007320.

			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.

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

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 *)

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A094683 Juggler sequence: if n mod 2 = 0 then floor(sqrt(n)) else floor(n^(3/2)).

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A007321 Number of steps needed for modified juggler sequence (A094685) 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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A094679 n sets a new record for number of iterations to reach 1 in the juggler sequence problem.

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A094685 Modified juggler sequence: if n mod 2 = 0 then round(sqrt(n)) else round(n^(3/2)).

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

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A095396 Modified juggler map: for even numbers: a(n) = floor(n^(2/3)) and for odd n: a(n) = floor(n^(3/2)) = floor(sqrt(n^3)).

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