A094683 -id:A094683

A007320

Number of steps needed for juggler sequence (A094683) started at n to reach 1.
(Formerly M4047)

+20
19

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

COMMENTS

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

REFERENCES

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).

LINKS

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

EXAMPLE

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.

MAPLE

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

MATHEMATICA

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 *)

CROSSREFS

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

KEYWORD

nonn

AUTHOR

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

EXTENSIONS

Corrected and extended by Jason Earls, Jun 09 2004

STATUS

approved

A094716

Largest value in trajectory of n under the juggler map of A094683.

+20
10

0, 1, 2, 36, 4, 36, 6, 18, 8, 140, 36, 36, 36, 46, 36, 58, 16, 70, 18, 140, 20, 140, 22, 110, 24, 52214, 36, 140, 36, 156, 36, 172, 36, 2598, 36, 2978, 36, 24906114455136, 38, 233046, 40, 262, 42, 4710, 44, 5222, 46, 322, 48, 6352, 50, 364, 52, 7554, 54, 8210, 56, 430, 58, 946636

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

Harry J. Smith found that the highest value in the trajectory of 30817 is a number with 45391 digits and the highest value in the trajectory of 48443 is a number with 972463 digits. - Jason Earls, Jun 10 2004

LINKS

Hans Havermann, Table of n, a(n) for n = 0..2188 (shortened by N. J. A. Sloane, Jan 13 2019)

Harry J. Smith, Juggler Numbers

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane, Jun 10 2004

EXTENSIONS

More terms from Jason Earls, Jun 10 2004

a(37) corrected by Labos Elemer, Jun 19 2004

a(49) corrected by Hans Havermann, Dec 07 2017

STATUS

approved

A094684

Records in A094683.

+20
6

0, 1, 5, 11, 18, 27, 36, 46, 58, 70, 82, 96, 110, 125, 140, 156, 172, 189, 207, 225, 243, 262, 281, 301, 322, 343, 364, 385, 407, 430, 453, 476, 500, 524, 548, 573, 598, 623, 649, 675, 702, 729, 756, 783, 811, 839, 868, 896, 925, 955, 985

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

Each odd n in A094683 sets a new record, so this is just a bisection of A094683.

LINKS

Table of n, a(n) for n=0..50.

CROSSREFS

Cf. A094683.

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Jun 09 2004

STATUS

approved

A218335

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

+20
1

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Arkadiusz Wesolowski, Table of n, a(n) for n = 1..10000

Eric Weisstein's World of Mathematics, Juggler Sequence

FORMULA

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

MATHEMATICA

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

CROSSREFS

Cf. A094683, A094716.

KEYWORD

nonn

AUTHOR

Arkadiusz Wesolowski, Oct 26 2012

STATUS

approved

A094804

Number of primes in the trajectory of n under the juggler map of A094683.

+20
0

0, 0, 1, 4, 1, 3, 1, 2, 1, 2, 4, 2, 4, 2, 4, 2, 1, 2, 1, 3, 1, 2, 1, 5, 1, 2, 3, 2, 3, 5, 3, 3, 3, 2, 3, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 1, 1, 3, 1, 4, 2, 3, 2, 3, 2, 2, 2, 1, 2, 4, 2, 3, 2, 1, 1, 1, 1, 6, 1, 4, 1, 2, 1, 3, 1, 1, 1, 3, 1, 4, 1, 1, 2, 3, 2, 4, 2, 3, 2, 3, 2, 5, 2, 5, 2, 2, 2, 4, 2, 1, 4, 3, 4, 2, 4

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,4

LINKS

Table of n, a(n) for n=0..104.

EXAMPLE

3 -> 5 -> 11 -> 36 -> 6 -> 2 -> 1, so in this trajectory 3, 5, 11 and 2 are primes, hence a(3) = 4.

MATHEMATICA

Table[Count[NestWhileList[If[EvenQ[#], Floor[Sqrt[#]], Floor[(Sqrt[#])^3]]&, n, #>1&], _?PrimeQ], {n, 0, 120}] (* Harvey P. Dale, Sep 26 2021 *)

KEYWORD

easy,nonn

AUTHOR

Jason Earls, Jun 11 2004

STATUS

approved

A094819

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

+20
0

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

LINKS

Table of n, a(n) for n=0..74.

CROSSREFS

Cf. A094683.

KEYWORD

nonn

AUTHOR

Jason Earls, Jun 12 2004

STATUS

approved

A007321

Number of steps needed for modified juggler sequence (A094685) started at n to reach 1.
(Formerly M4048)

+10
9

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

COMMENTS

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

REFERENCES

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).

LINKS

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

MAPLE

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

MATHEMATICA

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}]

CROSSREFS

Cf. A094685, A007320, A094683.

KEYWORD

nonn,easy

AUTHOR

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

EXTENSIONS

More terms from N. J. A. Sloane, Jun 09 2004

STATUS

approved

A094685

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

+10
5

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,4

REFERENCES

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

LINKS

Chai Wah Wu, Table of n, a(n) for n = 0..10000

H. J. Smith, Juggler Sequence

MAPLE

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

PROG

(Python)

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

CROSSREFS

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

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Jun 09 2004

STATUS

approved

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)).

+10
3

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

COMMENTS

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

LINKS

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

FORMULA

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

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; 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 *)

PROG

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

CROSSREFS

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

KEYWORD

nonn

AUTHOR

Labos Elemer, Jun 18 2004

EXTENSIONS

Name simplified by Antti Karttunen, May 28 2017

STATUS

approved

A095397

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

+10
1

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Parallel to A094716.

LINKS

Table of n, a(n) for n=1..58.

EXAMPLE

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

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}]

CROSSREFS

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

KEYWORD

nonn

AUTHOR

Labos Elemer, Jun 18 2004

STATUS

approved