A094716 -id:A094716 - 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

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

Original entry on oeis.org

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

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

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 *)

A218335 Even numbers n such that the largest value in trajectory of n under the juggler map of A094683 is greater than n.

Original entry on oeis.org

10, 12, 14, 26, 28, 30, 32, 34, 82, 84, 86, 88, 90, 92, 94, 96, 98, 626, 628, 630, 632, 634, 636, 638, 640, 642, 644, 646, 648, 650, 652, 654, 656, 658, 660, 662, 664, 666, 668, 670, 672, 674, 1090, 1092, 1094, 1096, 1098, 1100, 1102, 1104, 1106, 1108, 1110

Offset: 1

Arkadiusz Wesolowski, Oct 26 2012

nonn

Arkadiusz Wesolowski, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Juggler Sequence

Cf. A094683, A094716.

lst = {}; Do[n = a; While[True, If[EvenQ[a], a = Floor[a^(1/2)]; If[a == 1, Break[]], a = Floor[a^(3/2)]; If[a > n, AppendTo[lst, n]; Break[]]]], {a, 2, 1110, 2}]; lst

A number n is in the sequence if and only if n is even and A094716(n) > n.

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}]

cp's OEIS Frontend

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

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

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

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

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A218335 Even numbers n such that the largest value in trajectory of n under the juggler map of A094683 is greater than n.

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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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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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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}.