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# Juggler sequence

Juggler sequence
You are encouraged to solve this task according to the task description, using any language you may know.

Background of the   juggler sequence:

Juggler sequences were publicized by an American mathematician and author   Clifford A. Pickover.   The name of the sequence gets it's name from the similarity of the rising and falling nature of the numbers in the sequences,   much like balls in the hands of a juggler.

Description

A juggler sequence is an integer sequence that starts with a positive integer a[0], with each subsequent term in the sequence being defined by the recurrence relation:

```            a[k + 1]  =  floor(a[k] ^ 0.5)    if a[k] is even    or
a[k + 1]  =  floor(a[k] ^ 1.5)    if a[k] is odd
```

If a juggler sequence reaches 1, then all subsequent terms are equal to 1. This is known to be the case for initial terms up to 1,000,000 but it is not known whether all juggler sequences after that will eventually reach 1.

Compute and show here the following statistics for juggler sequences with an initial term of a[n] where n is between 20 and 39 inclusive:

• l[n] - the number of terms needed to reach 1.
• h[n] - the maximum value reached in that sequence.
• i[n] - the index of the term (starting from 0) at which the maximum is (first) reached.

If your language supports big integers with an integer square root function, also compute and show here the same statistics for as many as you reasonably can of the following values for n:

113, 173, 193, 2183, 11229, 15065, 15845, 30817, 48443, 275485, 1267909, 2264915, 5812827

Those with fast languages and fast machines may also like to try their luck at n = 7110201.

However, as h[n] for most of these numbers is thousands or millions of digits long, show instead of h[n]:

• d[n] - the number of digits in h[n]

The results can be (partially) verified against the table here.

• oeis:A007320 Number of steps needed for Juggler sequence started at n to reach 1
• oeis:A094716 Largest value in the Juggler sequence started at n

## 11l

Translation of: Nim
`F juggler(n)   V a = Int64(n)   V r_count = 0   V r_max = a   V r_maxidx = 0   L a != 1      V f = Float(a)      a = Int64(I a [&] 1 == 0 {sqrt(f)} E f * sqrt(f))      r_count++      I a > r_max         r_max = a         r_maxidx = r_count   R (r_count, r_max, r_maxidx) print(‘n   l[n]            h[n]  i[n]’)print(‘------------------------------’)L(n) 20..39   V (l, h, i) = juggler(n)   print(f:‘{n}   {l:2}  {h:14}     {i}’)`
Output:
```n   l[n]            h[n]  i[n]
------------------------------
20    3              20     0
21    9             140     4
22    3              22     0
23    9             110     1
24    3              24     0
25   11           52214     3
26    6              36     3
27    6             140     1
28    6              36     3
29    9             156     1
30    6              36     3
31    6             172     1
32    6              36     3
33    8            2598     2
34    6              36     3
35    8            2978     2
36    3              36     0
37   17  24906114455136     8
38    3              38     0
39   14          233046     3
```
Translation of: Python
`F isqrt(BigInt x)   assert(x >= 0)    V q = BigInt(1)   L q <= x      q *= 4    V z = x   V r = BigInt(0)   L q > 1      q I/= 4      V t = z - r - q      r I/= 2      I t >= 0         z = t         r += q    R r F juggler(k, countdig = 1B, maxiters = 1000)   V (m, maxj, maxjpos) = (BigInt(k), BigInt(k), 0)   L(i) 1 .< maxiters      m = I m % 2 == 0 {isqrt(m)} E isqrt(m * m * m)      I m >= maxj         (maxj, maxjpos) = (m, i)      I m == 1         print(f:‘{k:9}{commatize(i):6}{maxjpos:6}{commatize(I countdig {String(maxj).len} E maxj):20}{I countdig {‘ digits’} E ‘’}’)         R i    print(‘ERROR: Juggler series starting with ’k‘ did not converge in ’maxiters‘ iterations’) print("       n    l(n)  i(n)       h(n) or d(n)\n-------------------------------------------")L(k) 20..39   juggler(k, 0B) L(k) [113, 173, 193, 2183, 11229, 15065]   juggler(k)`
Output:
```       n    l(n)  i(n)       h(n) or d(n)
-------------------------------------------
20     3     0                  20
21     9     4                 140
22     3     0                  22
23     9     1                 110
24     3     0                  24
25    11     3              52,214
26     6     3                  36
27     6     1                 140
28     6     3                  36
29     9     1                 156
30     6     3                  36
31     6     1                 172
32     6     3                  36
33     8     2               2,598
34     6     3                  36
35     8     2               2,978
36     3     0                  36
37    17     8  24,906,114,455,136
38     3     0                  38
39    14     3             233,046
113    16     9                  27 digits
173    32    17                  82 digits
193    73    47                 271 digits
2183    72    32               5,929 digits
11229   101    54               8,201 digits
15065    66    25              11,723 digits
```

`with Ada.Text_IO;with Ada.Numerics.Generic_Elementary_Functions; procedure Juggler is    subtype Number  is Long_Long_Integer;   type Index_Type is new Natural;    subtype Initial_Values is Number range 20 .. 39;    generic      Initial : Number;   package Generic_Juggler is      procedure Next (Value : out Number; Index : out Index_Type);   end Generic_Juggler;    package body Generic_Juggler is       type Real is new Long_Long_Float;       package Real_Math is        new Ada.Numerics.Generic_Elementary_Functions (Real);       K   : Index_Type := 0;      A_K : Real       := Real (Initial);       procedure Next (Value : out Number; Index : out Index_Type) is         use Real_Math;      begin         Value := Number (A_K);         Index := K;         A_K := (if Number (A_K) mod 2 = 0                 then Real'Floor (A_K ** 0.5)                 else Real'Floor (A_K ** 1.5));         K := K + 1;      end Next;    end Generic_Juggler;    procedure Statistics (N   : Number;     L_N : out Index_Type;                         H_N : out Number; I_N : out Index_Type)   is      package Juggler_Generator is new Generic_Juggler (Initial => N);      use Juggler_Generator;      Value : Number;   begin      H_N := 0;      I_N := 0;      loop         Next (Value, L_N);         if Value > H_N then            H_N := Value;            I_N := L_N;         end if;         exit when Value = 1;      end loop;   end Statistics;    procedure Put_Table is      package Number_IO is new Ada.Text_IO.Integer_IO (Number);      package Index_IO  is new Ada.Text_IO.Integer_IO (Index_Type);      use Ada.Text_IO, Number_IO, Index_IO;      L_N : Index_Type;      H_N : Number;      I_N : Index_Type;   begin      Put_Line ("  N   L(N)            H(N)   I(N)");      Put_Line ("---------------------------------");      for N in Initial_Values loop         Statistics (N, L_N, H_N, I_N);         Put (N, Width => 3);     Put (L_N, Width => 7);         Put (H_N, Width => 16);  Put (I_N, Width => 7);  New_Line;      end loop;   end Put_Table; begin   Put_Table;end Juggler;`
Output:
```  N   L(N)            H(N)   I(N)
---------------------------------
20      3              20      0
21      9             140      4
22      3              22      0
23      9             110      1
24      3              24      0
25     11           52214      3
26      6              36      3
27      6             140      1
28      6              36      3
29      9             156      1
30      6              36      3
31      6             172      1
32      6              36      3
33      8            2598      2
34      6              36      3
35      8            2978      2
36      3              36      0
37     17  24906114455136      8
38      3              38      0
39     14          233046      3
```

## AppleScript

### Core language

Keeping within AppleScript's usable number range:

`on juggler(n)    script o        property sequence : {n}    end script     set i to 1    set {max, pos} to {n, i}    repeat until (n = 1)        set n to n ^ (n mod 2 + 0.5) div 1        set end of o's sequence to n        set i to i + 1        if (n > max) then set {max, pos} to {n, i}    end repeat     return {n:n, sequence:o's sequence, |length|:i, max:max, maxPos:pos}end juggler on intToText(n)    if (n < 2 ^ 29) then return n as integer as text    set lst to {n mod 10 as integer}    set n to n div 10    repeat until (n = 0)        set beginning of lst to n mod 10 as integer        set n to n div 10    end repeat     return join(lst, "")end intToText on join(lst, delim)    set astid to AppleScript's text item delimiters    set AppleScript's text item delimiters to delim    set txt to lst as text    set AppleScript's text item delimiters to astid    return txtend join on task()    set output to {}    repeat with n from 20 to 39        set {|length|:len, max:max, maxPos:pos} to juggler(n)        set end of output to join({n, ": l[n] = ", len - 1, ", h[n] = ", intToText(max), ", i[n] = ", pos - 1}, "")    end repeat     return join(output, linefeed)end task task()`
Output:
`"20: l[n] = 3, h[n] = 20, i[n] = 021: l[n] = 9, h[n] = 140, i[n] = 422: l[n] = 3, h[n] = 22, i[n] = 023: l[n] = 9, h[n] = 110, i[n] = 124: l[n] = 3, h[n] = 24, i[n] = 025: l[n] = 11, h[n] = 52214, i[n] = 326: l[n] = 6, h[n] = 36, i[n] = 327: l[n] = 6, h[n] = 140, i[n] = 128: l[n] = 6, h[n] = 36, i[n] = 329: l[n] = 9, h[n] = 156, i[n] = 130: l[n] = 6, h[n] = 36, i[n] = 331: l[n] = 6, h[n] = 172, i[n] = 132: l[n] = 6, h[n] = 36, i[n] = 333: l[n] = 8, h[n] = 2598, i[n] = 234: l[n] = 6, h[n] = 36, i[n] = 335: l[n] = 8, h[n] = 2978, i[n] = 236: l[n] = 3, h[n] = 36, i[n] = 037: l[n] = 17, h[n] = 24906114455136, i[n] = 838: l[n] = 3, h[n] = 38, i[n] = 039: l[n] = 14, h[n] = 233046, i[n] = 3"`

### Shell script

One of AppleScript's main roles is telling other software to do things. This includes Unix executables, many of which come with the system. In the following, the 'do shell script' command feeds a script to the Bash shell, which script itself contains code to be passed to and executed by the "bc" executable. It's essentially a script within a script within a script. The text returned from "bc", which can handle larger numbers than core AppleScript, contains lines which are just the zeros returned by the 'juggler' function, so these are stripped out using "sed". The 'do shell script' command is supplied by the StandardAdditions OSAX which comes with the system as a standard AppleScript extension. So ironically, there's not a single command from the core language in the following code. But it's legitimate AppleScript and the input and output are both AppleScript text objects.

`do shell script "echo 'define juggler(n) {    #auto temp,i,max,pos    #scale = 0;    temp = n; i = 0; max = n; pos = i;    while (temp > 1) {        i = i + 1; temp = sqrt(temp ^ (1 + (temp % 2 * 2)));         if (temp > max) { max = temp; pos = i; }    }    if (n < 40) {        print n,\": l[n] = \",i,\", h[n] = \",max, \", i[n] = \",pos,\"\\n\";    } else {        print n,\": l[n] = \",i,\", d[n] = \",length(max), \", i[n] = \",pos,\"\\n\";    }    return;} for (n = 20 ; n < 40 ; n++) { juggler(n); }print \"\\n\";juggler(113); juggler(173); juggler(193); juggler(2183); juggler(11229); juggler(15065);juggler(15845); # 91 seconds so far.juggler(30817); # Another 191 to here.# juggler(48443) produced no result after running all night.' | bc | sed -n '/^0\$/ !p;'"`
Output:
`"20: l[n] = 3, h[n] = 20, i[n] = 021: l[n] = 9, h[n] = 140, i[n] = 422: l[n] = 3, h[n] = 22, i[n] = 023: l[n] = 9, h[n] = 110, i[n] = 124: l[n] = 3, h[n] = 24, i[n] = 025: l[n] = 11, h[n] = 52214, i[n] = 326: l[n] = 6, h[n] = 36, i[n] = 327: l[n] = 6, h[n] = 140, i[n] = 128: l[n] = 6, h[n] = 36, i[n] = 329: l[n] = 9, h[n] = 156, i[n] = 130: l[n] = 6, h[n] = 36, i[n] = 331: l[n] = 6, h[n] = 172, i[n] = 132: l[n] = 6, h[n] = 36, i[n] = 333: l[n] = 8, h[n] = 2598, i[n] = 234: l[n] = 6, h[n] = 36, i[n] = 335: l[n] = 8, h[n] = 2978, i[n] = 236: l[n] = 3, h[n] = 36, i[n] = 037: l[n] = 17, h[n] = 24906114455136, i[n] = 838: l[n] = 3, h[n] = 38, i[n] = 039: l[n] = 14, h[n] = 233046, i[n] = 3 113: l[n] = 16, d[n] = 27, i[n] = 9173: l[n] = 32, d[n] = 82, i[n] = 17193: l[n] = 73, d[n] = 271, i[n] = 472183: l[n] = 72, d[n] = 5929, i[n] = 3211229: l[n] = 101, d[n] = 8201, i[n] = 5415065: l[n] = 66, d[n] = 11723, i[n] = 2515845: l[n] = 139, d[n] = 23889, i[n] = 4330817: l[n] = 93, d[n] = 45391, i[n] = 39"`

## BQN

`Juggle ← {    Step ← ⌊⊢⋆(0.5 + 2|⊢)     ¯1‿0‿0 + 3↑{           n‿imax‿max‿term ← 𝕩        𝕊⍟(term≠1) ⟨n+1, (max<term)⊑imax‿n, max⌈term, Step term⟩    } 0‿0‿0‿𝕩} >⟨"NLIH"⟩ ∾ (⊢∾Juggle)¨ 20+↕20`
Output:
```┌─
╵ 'N' 'L' 'I' 'H'
20  3   0   20
21  9   4   140
22  3   0   22
23  9   1   110
24  3   0   24
25  11  3   52214
26  6   3   36
27  6   1   140
28  6   3   36
29  9   1   156
30  6   3   36
31  6   1   172
32  6   3   36
33  8   2   2598
34  6   3   36
35  8   2   2978
36  3   0   36
37  17  8   24906114455136
38  3   0   38
39  14  3   233046
┘```

## C++

Translation of: Go
Library: GMP
`#include <cassert>#include <iomanip>#include <iostream>#include <string> #include <gmpxx.h> using big_int = mpz_class; auto juggler(int n) {    assert(n >= 1);    int count = 0, max_count = 0;    big_int a = n, max = n;    while (a != 1) {        if (a % 2 == 0)            a = sqrt(a);        else            a = sqrt(big_int(a * a * a));        ++count;        if (a > max) {            max = a;            max_count = count;        }    }    return std::make_tuple(count, max_count, max, max.get_str().size());} int main() {    std::cout.imbue(std::locale(""));    std::cout << "n    l[n]  i[n]   h[n]\n";    std::cout << "--------------------------------\n";    for (int n = 20; n < 40; ++n) {        auto [count, max_count, max, digits] = juggler(n);        std::cout << std::setw(2) << n << "    " << std::setw(2) << count                  << "    " << std::setw(2) << max_count << "    " << max                  << '\n';    }    std::cout << '\n';    std::cout << "       n       l[n]   i[n]   d[n]\n";    std::cout << "----------------------------------------\n";    for (int n : {113, 173, 193, 2183, 11229, 15065, 15845, 30817, 48443,                  275485, 1267909, 2264915, 5812827, 7110201, 56261531,                  92502777, 172376627, 604398963}) {        auto [count, max_count, max, digits] = juggler(n);        std::cout << std::setw(11) << n << "    " << std::setw(3) << count                  << "    " << std::setw(3) << max_count << "    " << digits                  << '\n';    }}`
Output:
```n    l[n]  i[n]   h[n]
--------------------------------
20     3     0    20
21     9     4    140
22     3     0    22
23     9     1    110
24     3     0    24
25    11     3    52214
26     6     3    36
27     6     1    140
28     6     3    36
29     9     1    156
30     6     3    36
31     6     1    172
32     6     3    36
33     8     2    2598
34     6     3    36
35     8     2    2978
36     3     0    36
37    17     8    24906114455136
38     3     0    38
39    14     3    233046

n       l[n]   i[n]   d[n]
----------------------------------------
113     16      9    27
173     32     17    82
193     73     47    271
2,183     72     32    5,929
11,229    101     54    8,201
15,065     66     25    11,723
15,845    139     43    23,889
30,817     93     39    45,391
48,443    157     60    972,463
275,485    225    148    1,909,410
1,267,909    151     99    1,952,329
2,264,915    149     89    2,855,584
5,812,827    135     67    7,996,276
7,110,201    205    119    89,981,517
56,261,531    254     92    105,780,485
92,502,777    191    117    139,486,096
172,376,627    262     90    449,669,621
604,398,963    327    172    640,556,693
```

## F#

` // Juggler sequence. Nigel Galloway: August 19th., 2021let J n=Seq.unfold(fun(n,i,g,l)->if n=1I then None else let e=match n.IsEven with true->Isqrt n |_->Isqrt(n**3) in Some((i,g,l),if e>i then (e,e,l+1,l+1) else (e,i,g,l+1)))(n,n,0,0)|>Seq.lastprintfn " n  l[n] i[n]  h[n]\n___________________"; [20I..39I]|>Seq.iter(fun n->let i,g,l=J n in printfn \$"%d{int n}%5d{l+1}%5d{g}   %A{i}")printfn "      n  l[n] i[n]  d[n]\n________________________"; [113I;173I;193I;2183I;11229I;15065I;15845I;30817I]|>Seq.iter(fun n->let i,g,l=J n in printfn \$"%8d{int n}%5d{l+1}%5d{g}   %d{(bigint.Log10>>int>>(+)1) i}") `
Output:
``` n  l[n] i[n]  h[n]
___________________
20    3    0   20
21    9    4   140
22    3    0   22
23    9    1   110
24    3    0   24
25   11    3   52214
26    6    3   36
27    6    1   140
28    6    3   36
29    9    1   156
30    6    3   36
31    6    1   172
32    6    3   36
33    8    2   2598
34    6    3   36
35    8    2   2978
36    3    0   36
37   17    8   24906114455136
38    3    0   38
39   14    3   233046

n  l[n] i[n]  d[n]
________________________
113   16    9   27
173   32   17   82
193   73   47   271
2183   72   32   5929
11229  101   54   8201
15065   66   25   11723
15845  139   43   23889
30817   93   39   45391
```

## Factor

Works with: Factor version 0.99 2021-06-02
`USING: combinators formatting generalizations io kernel mathmath.extras math.functions.integer-logs math.order math.rangessequences strings tools.memory.private ; : next ( m -- n )    dup odd? [ dup dup * * ] when integer-sqrt ; : new-max ( l i h a -- l i h a )    [ drop dup ] 2dip nip dup ; : (step) ( l i h a -- l i h a )    [ 1 + ] 3dip 2dup < [ new-max ] when next ; : step ( l i h a -- l i h a )    dup 1 = [ (step) ] unless ; : juggler ( n quot: ( h -- obj ) -- l i h )    [ 0 0 ] [ dup [ step ] to-fixed-point drop ] [ call ] tri*    [ 1 [-] ] dip ; inline CONSTANT: fmt "%-8s %-8s %-8s %s\n" : row. ( n quot -- )    dupd juggler [ commas ] 4 napply fmt printf ; inline : dashes. ( n -- )    CHAR: - <string> print ; : header. ( str -- )    [ "n" "l[n]" "i[n]" ] dip fmt printf 45 dashes. ; : juggler. ( seq quot str -- )    header. [ row. ] curry each ; inline 20 39 [a,b] [ ] "h[n]" juggler. nl { 113 173 193 2183 11229 15065 15845 30817 }[ integer-log10 1 + ] "d[n]" juggler.`
Output:
```n        l[n]     i[n]     h[n]
---------------------------------------------
20       3        0        20
21       9        4        140
22       3        0        22
23       9        1        110
24       3        0        24
25       11       3        52,214
26       6        3        36
27       6        1        140
28       6        3        36
29       9        1        156
30       6        3        36
31       6        1        172
32       6        3        36
33       8        2        2,598
34       6        3        36
35       8        2        2,978
36       3        0        36
37       17       8        24,906,114,455,136
38       3        0        38
39       14       3        233,046

n        l[n]     i[n]     d[n]
---------------------------------------------
113      16       9        27
173      32       17       82
193      73       47       271
2,183    72       32       5,929
11,229   101      54       8,201
15,065   66       25       11,723
15,845   139      43       23,889
30,817   93       39       45,391
```

## Go

Translation of: Wren
Library: Go-rcu

This originally took about 13.5 minutes to reach n = 5,812,827 on my machine (Intel core i7-8565U) using Go's native 'math/big' package.

However, when I exchanged that for Go's GMP wrapper there was a massive speed-up (now only 6.4 seconds to reach n = 5,812,827) and even 7,110,201 became viable with an overall time of 1 minute 40 seconds.

The next four record holders for the largest term (see talk page), are also doable but increased the overall time to nearly 24 minutes on my machine.

`package main import (    "fmt"    "log"    //"math/big"    big "github.com/ncw/gmp"    "rcu") var zero = new(big.Int)var one = big.NewInt(1)var two = big.NewInt(2) func juggler(n int64) (int, int, *big.Int, int) {    if n < 1 {        log.Fatal("Starting value must be a positive integer.")    }    count := 0    maxCount := 0    a := big.NewInt(n)    max := big.NewInt(n)    tmp := new(big.Int)    for a.Cmp(one) != 0 {        if tmp.Rem(a, two).Cmp(zero) == 0 {            a.Sqrt(a)        } else {            tmp.Mul(a, a)            tmp.Mul(tmp, a)            a.Sqrt(tmp)        }        count++        if a.Cmp(max) > 0 {            max.Set(a)            maxCount = count        }    }    return count, maxCount, max, len(max.String())} func main() {    fmt.Println("n    l[n]  i[n]  h[n]")    fmt.Println("-----------------------------------")    for n := int64(20); n < 40; n++ {        count, maxCount, max, _ := juggler(n)        cmax := rcu.Commatize(int(max.Int64()))        fmt.Printf("%2d    %2d   %2d    %s\n", n, count, maxCount, cmax)    }    fmt.Println()    nums := []int64{        113, 173, 193, 2183, 11229, 15065, 15845, 30817, 48443, 275485, 1267909,        2264915, 5812827, 7110201, 56261531, 92502777, 172376627, 604398963,    }    fmt.Println("       n      l[n]   i[n]   d[n]")    fmt.Println("-------------------------------------")    for _, n := range nums {        count, maxCount, _, digits := juggler(n)        cn := rcu.Commatize(int(n))        fmt.Printf("%11s   %3d    %3d    %s\n", cn, count, maxCount, rcu.Commatize(digits))    }}`
Output:
```n    l[n]  i[n]  h[n]
-----------------------------------
20     3    0    20
21     9    4    140
22     3    0    22
23     9    1    110
24     3    0    24
25    11    3    52,214
26     6    3    36
27     6    1    140
28     6    3    36
29     9    1    156
30     6    3    36
31     6    1    172
32     6    3    36
33     8    2    2,598
34     6    3    36
35     8    2    2,978
36     3    0    36
37    17    8    24,906,114,455,136
38     3    0    38
39    14    3    233,046

n      l[n]   i[n]   d[n]
-------------------------------------
113    16      9    27
173    32     17    82
193    73     47    271
2,183    72     32    5,929
11,229   101     54    8,201
15,065    66     25    11,723
15,845   139     43    23,889
30,817    93     39    45,391
48,443   157     60    972,463
275,485   225    148    1,909,410
1,267,909   151     99    1,952,329
2,264,915   149     89    2,855,584
5,812,827   135     67    7,996,276
7,110,201   205    119    89,981,517
56,261,531   254     92    105,780,485
92,502,777   191    117    139,486,096
172,376,627   262     90    449,669,621
604,398,963   327    172    640,556,693
```

Integer square root is computed as in Isqrt_(integer_square_root)_of_X#Haskell

`import Text.Printfimport Data.List juggler :: Integer -> [Integer]juggler = takeWhile (> 1) . iterate (\x -> if odd x                                           then isqrt (x*x*x)                                           else isqrt x) task :: Integer -> IO ()task n = printf s n (length ns + 1) (i :: Int) (showMax m)  where    ns = juggler n    (m, i) = maximum \$ zip ns [0..]    s = "n = %d length = %d maximal value at = %d (%s)\n"    showMax n = let s = show n                in if n > 10^100                   then show (length s) ++ " digits"                    else show n  main = do  mapM_ task [20..39]  putStrLn "\nTough guys\n"  mapM_ task [ 113, 173, 193, 2183, 11229, 15065, 15845, 30817 ]`
```n = 20 length = 3 maximal value at = 0 (20)
n = 21 length = 10 maximal value at = 4 (140)
n = 22 length = 3 maximal value at = 0 (22)
n = 23 length = 10 maximal value at = 1 (110)
n = 24 length = 3 maximal value at = 0 (24)
n = 25 length = 11 maximal value at = 3 (52214)
n = 26 length = 7 maximal value at = 3 (36)
n = 27 length = 7 maximal value at = 1 (140)
n = 28 length = 7 maximal value at = 3 (36)
n = 29 length = 10 maximal value at = 1 (156)
n = 30 length = 7 maximal value at = 3 (36)
n = 31 length = 7 maximal value at = 1 (172)
n = 32 length = 7 maximal value at = 3 (36)
n = 33 length = 8 maximal value at = 2 (2598)
n = 34 length = 7 maximal value at = 3 (36)
n = 35 length = 8 maximal value at = 2 (2978)
n = 36 length = 4 maximal value at = 0 (36)
n = 37 length = 18 maximal value at = 8 (24906114455136)
n = 38 length = 4 maximal value at = 0 (38)
n = 39 length = 15 maximal value at = 3 (233046)

Tough guys

n = 113 length = 17 maximal value at = 9 (202924588924125339424550328)
n = 173 length = 33 maximal value at = 17 (4450608860210678234719664930918817118564659064289879586228390154864378511410864886)
n = 193 length = 74 maximal value at = 47 (271 digits)
n = 2183 length = 73 maximal value at = 32 (5929 digits)
n = 11229 length = 102 maximal value at = 54 (8201 digits)
n = 15065 length = 67 maximal value at = 25 (11723 digits)
n = 15845 length = 140 maximal value at = 43 (23889 digits)
n = 30817 length = 94 maximal value at = 39 (45391 digits)```

## J

`jug=: <[email protected]^ 0.5+2|]`
would work if 64 bit floats were adequate for the task example, but they are not.

Instead, we take the square root of either the even number or the third power of the odd number:

`jugx=: <[email protected]%:@(^ 1x+2*2|])`

`require'format/printf' task=: {{  echo '%d: l: %d, h: %d, i:%d' sprintf y;(#;>./;]i.>./)jugx^:a: y}}    task"0(+i.)2020: l: 4, h: 20, i:021: l: 10, h: 140, i:422: l: 4, h: 22, i:023: l: 10, h: 110, i:124: l: 4, h: 24, i:025: l: 12, h: 52214, i:326: l: 7, h: 36, i:327: l: 7, h: 140, i:128: l: 7, h: 36, i:329: l: 10, h: 156, i:130: l: 7, h: 36, i:331: l: 7, h: 172, i:132: l: 7, h: 36, i:333: l: 9, h: 2598, i:234: l: 7, h: 36, i:335: l: 9, h: 2978, i:236: l: 4, h: 36, i:037: l: 18, h: 24906114455136, i:838: l: 4, h: 38, i:039: l: 15, h: 233046, i:3`

Sadly, J's extended precision implementation is antiquated (slow), hopefully that will be fixed before too long.

Still, some of the stretch exercises can be computed quickly:

`taskx=: {{  echo '%d: l: %d, d: %d, i:%d' sprintf y;(#;#@":@(>./);]i.>./)jugx^:a: y}}    taskx"0(113 173 193 2183 11229)113: l: 17, d: 27, i:9173: l: 33, d: 82, i:17193: l: 74, d: 271, i:472183: l: 73, d: 5929, i:3211229: l: 102, d: 8201, i:54`

## jq

Translation of: Wren

Works with gojq, the Go implementation of jq

The following jq program uses `idivide/1`, `isqrt/0`, and `lpad/1` as defined at Isqrt_(integer_square_root)_of_X#jq.

`def juggler:  . as \$n  | if \$n < 1 then "juggler starting value must be a positive integer." | error    else { a: \$n, count: 0, maxCount: 0, max: \$n }    | until (.a == 1;        if .a % 2 == 0 then .a |= isqrt        else .a = ((.a * .a * .a)|isqrt)        end        | .count += 1        | if .a > .max          then .max = .a          | .maxCount = .count	  else .	  end)    | [.count, .maxCount, .max, (.max|tostring|length)]    end; def fmt(a;b;c;d):  "\(.[0]|lpad(a)) \(.[1]|lpad(b)) \(.[2]|lpad(c)) \(.[3]|lpad(d))" ; def task1:  "n    l[n]  i[n]                h[n]",  "-----------------------------------",  (range(20; 40)   | . as \$n   | juggler as \$res   | [\$n, \$res[0], \$res[1], \$res[2] ]   | fmt(4;4;4;14) ) ; def task2:  def nums:[113, 173, 193, 2183, 11229, 15065, 15845, 30817];  "   n     l[n]   i[n]     d[n]",  "-----------------------------",  (nums[]   | . as \$n   | juggler as \$res   | [\$n, \$res[0], \$res[1], \$res[3] ]   | fmt(6; 6; 6; 8) ); task1, "", task2`
Output:
```n    l[n]  i[n]                h[n]
-----------------------------------
20    3    0             20
21    9    4            140
22    3    0             22
23    9    1            110
24    3    0             24
25   11    3          52214
26    6    3             36
27    6    1            140
28    6    3             36
29    9    1            156
30    6    3             36
31    6    1            172
32    6    3             36
33    8    2           2598
34    6    3             36
35    8    2           2978
36    3    0             36
37   17    8 24906114455136
38    3    0             38
39   14    3         233046

n     l[n]   i[n]     d[n]
-----------------------------
113     16      9       27
173     32     17       82
193     73     47      271
2183     72     32     5929
11229    101     54     8201
15065     66     25    11723
15845    139     43    23889
30817     93     39    45391
```

## Julia

`using Formatting function juggler(k, countdig=true, maxiters=20000)    m, maxj, maxjpos = BigInt(k), BigInt(k), BigInt(0)    for i in 1:maxiters        m = iseven(m) ? isqrt(m) : isqrt(m*m*m)        if m >= maxj            maxj, maxjpos  = m, i        end        if m == 1            println(lpad(k, 9), lpad(i, 6), lpad(maxjpos, 6), lpad(format(countdig ?                ndigits(maxj) : Int(maxj), commas=true), 20), countdig ? " digits" : "")            return i        end    end    error("Juggler series starting with \$k did not converge in \$maxiters iterations")end println("       n    l(n)  i(n)       h(n) or d(n)\n------------------------------------------")foreach(k -> juggler(k, false), 20:39)@time foreach(juggler,    [113, 173, 193, 2183, 11229, 15065, 15845, 30817, 48443, 275485, 1267909,     2264915, 5812827])@time juggler(7110201) `
Output:
```       n    l(n)  i(n)       h(n) or d(n)
------------------------------------------
20     3     0                  20
21     9     4                 140
22     3     0                  22
23     9     1                 110
24     3     0                  24
25    11     3              52,214
26     6     3                  36
27     6     1                 140
28     6     3                  36
29     9     1                 156
30     6     3                  36
31     6     1                 172
32     6     3                  36
33     8     2               2,598
34     6     3                  36
35     8     2               2,978
36     3     0                  36
37    17     8  24,906,114,455,136
38     3     0                  38
39    14     3             233,046
113    16     9                  27 digits
173    32    17                  82 digits
193    73    47                 271 digits
2183    72    32               5,929 digits
11229   101    54               8,201 digits
15065    66    25              11,723 digits
15845   139    43              23,889 digits
30817    93    39              45,391 digits
48443   157    60             972,463 digits
275485   225   148           1,909,410 digits
1267909   151    99           1,952,329 digits
2264915   149    89           2,855,584 digits
5812827   135    67           7,996,276 digits
4.969021 seconds (113.27 k allocations: 2.023 GiB, 0.29% gc time)
7110201   205   119          89,981,517 digits
89.493898 seconds (1.11 M allocations: 27.713 GiB, 1.19% gc time)
```

## Mathematica / Wolfram Language

`next[n_Integer] := If[[email protected], Floor[Sqrt[n]], Floor[n^(3/2)]] stats[n_Integer] :=  Block[{data = [email protected][next, n, # > 1 &], mx},  mx = [email protected][data, -1];  {n, Length[data], data[[mx]], mx - 1}] {TableForm[Table[[email protected], {n, 20, 39}],  TableHeadings -> {None, {"n", "length", "max", "max pos"}}]`
Output:
```
n	length	max	max pos
20	3	20	0
21	9	140	4
22	3	22	0
23	9	110	1
24	3	24	0
25	11	52214	3
26	6	36	3
27	6	140	1
28	6	36	3
29	9	156	1
30	6	36	3
31	6	172	1
32	6	36	3
33	8	2598	2
34	6	36	3
35	8	2978	2
36	3	36	0
37	17	24906114455136	8
38	3	38	0
39	14	233046	3

```

## Nim

Using only standard library, so limited to values of `n` less than 40.

`import math, strformat func juggler(n: Positive): tuple[count: int; max: uint64; maxIdx: int] =  var a = n.uint64  result = (0, a, 0)  while a != 1:    let f = float(a)    a = if (a and 1) == 0: sqrt(f).uint64        else: uint64(f * sqrt(f))    inc result.count    if a > result.max:      result.max = a      result.maxIdx = result.count echo "n   l[n]            h[n]  i[n]"echo "——————————————————————————————"for n in 20..39:  let (l, h, i) = juggler(n)  echo &"{n}   {l:2}  {h:14}     {i}"`
Output:
```n   l[n]            h[n]  i[n]
——————————————————————————————
20    3              20     0
21    9             140     4
22    3              22     0
23    9             110     1
24    3              24     0
25   11           52214     3
26    6              36     3
27    6             140     1
28    6              36     3
29    9             156     1
30    6              36     3
31    6             172     1
32    6              36     3
33    8            2598     2
34    6              36     3
35    8            2978     2
36    3              36     0
37   17  24906114455136     8
38    3              38     0
39   14          233046     3```

## Perl

`#!/usr/bin/perl use strict; # https://rosettacode.org/wiki/Juggler_sequenceuse warnings;use Math::BigInt lib => 'GMP'; print "       n  l(n) i(n)  h(n) or d(n)\n";print " -------  ---- ----  ------------\n";for my \$i ( 20 .. 39,  113, 173, 193, 2183, 11229, 15065, 15845, 30817,  48443, 275485, 1267909, 2264915, 5812827,  7110201  # tried my luck, luck takes about 94 seconds  )  {  my \$max = my \$n = Math::BigInt->new(\$i);  my \$at = my \$count = 0;  while( \$n > 1 )    {    \$n = sqrt( \$n & 1 ? \$n ** 3 : \$n );    \$count++;    \$n > \$max and (\$max, \$at) = (\$n, \$count);    }   if( length \$max < 27 )    {    printf "%8d  %4d  %3d  %s\n", \$i, \$count, \$at, \$max;    }  else    {    printf "%8d  %4d  %3d  d(n) = %d digits\n", \$i, \$count, \$at, length \$max;    }  }`
Output:
```       n  l(n) i(n)  h(n) or d(n)
-------  ---- ----  ------------
20     3    0  20
21     9    4  140
22     3    0  22
23     9    1  110
24     3    0  24
25    11    3  52214
26     6    3  36
27     6    1  140
28     6    3  36
29     9    1  156
30     6    3  36
31     6    1  172
32     6    3  36
33     8    2  2598
34     6    3  36
35     8    2  2978
36     3    0  36
37    17    8  24906114455136
38     3    0  38
39    14    3  233046
113    16    9  d(n) = 27 digits
173    32   17  d(n) = 82 digits
193    73   47  d(n) = 271 digits
2183    72   32  d(n) = 5929 digits
11229   101   54  d(n) = 8201 digits
15065    66   25  d(n) = 11723 digits
15845   139   43  d(n) = 23889 digits
30817    93   39  d(n) = 45391 digits
48443   157   60  d(n) = 972463 digits
275485   225  148  d(n) = 1909410 digits
1267909   151   99  d(n) = 1952329 digits
2264915   149   89  d(n) = 2855584 digits
5812827   135   67  d(n) = 7996276 digits
7110201   205  119  d(n) = 89981517 digits
```

## Phix

Library: Phix/online

You can run this online here.

```with javascript_semantics
include mpfr.e

function juggler(integer n, bool bDigits=false)
atom t0 = time(), t1 = time()+1
assert(n>=1)
mpz a = mpz_init(n),
h = mpz_init(n)
integer l = 0,
hi = 0
while mpz_cmp_si(a,1)!=0 do
if mpz_odd(a) then
mpz_pow_ui(a,a,3)
end if
mpz_sqrt(a,a)
l += 1
if mpz_cmp(a,h)>0 then
mpz_set(h,a)
hi = l
end if
if platform()!=JS and time()>t1 then
progress("working (l=%d)\r",{l})
t1 = time()+1
end if
end while
atom hd = iff(bDigits?mpz_sizeinbase(h,10):mpz_get_atom(h))
t0 = time()-t0
string t = iff(t0>=1?" ("&elapsed(t0)&")":"")
return {n, l, hd, hi, t}
end function

procedure main()
atom t0 = time()
printf(1," n  l[n]                 h[n]  i[n]\n")
printf(1,"-----------------------------------\n")
for n=20 to 39 do
printf(1,"%2d    %2d   %,18d    %2d %s\n", juggler(n))
end for
sequence nums = {113, 173, 193, 2183, 11229, 15065, 15845, 30817}
-- alas mpz_sqrt() throws a wobbly over the rest:
--                   48443, 275485, 1267909, 2264915, 5812827, 7110201}
printf(1,"\n        n  l[n]    d[n]  i[n]\n")
printf(1,"-----------------------------\n")
--  ... and pwa/p2js(/mpfr.js) only copes with the first 3
for i=1 to iff(platform()=JS?3:length(nums)) do
printf(1,"%9d   %3d  %,6d    %2d %s\n", juggler(nums[i],true))
end for
?elapsed(time()-t0)
end procedure
main()
```
Output:
``` n  l[n]                 h[n]  i[n]
-----------------------------------
20     3                   20     0
21     9                  140     4
22     3                   22     0
23     9                  110     1
24     3                   24     0
25    11               52,214     3
26     6                   36     3
27     6                  140     1
28     6                   36     3
29     9                  156     1
30     6                   36     3
31     6                  172     1
32     6                   36     3
33     8                2,598     2
34     6                   36     3
35     8                2,978     2
36     3                   36     0
37    17   24,906,114,455,136     8
38     3                   38     0
39    14              233,046     3

n  l[n]    d[n]  i[n]
-----------------------------
113    16      27     9
173    32      82    17
193    73     271    47
2183    72   5,929    32
11229   101   8,201    54
15065    66  11,723    25
15845   139  23,889    43
30817    93  45,391    39
"0.1s"
```

Actually pwa/p2js will cope a bit further than that 193, but at a pretty glacial rate (your browser's JavaScript BigInt implementation/performance might differ from mine):

```     2183    72   5,929    32  (1 minute and 8s)
11229   101   8,201    54  (2 minutes and 33s)
15065    66  11,723    25  (7 minutes and 43s)
15845   139  23,889    43  (1 hour, 10 minutes and 32s)
"1 hour, 21 minutes and 47s"
```

(Everything prior was over and done in 0.1s) I think I can fairly safely predict that 30817 would be over 4 hours. Should someone provide a better implementation of mpz_sqrt() for either or both of desktop/Phix and pwa/p2js's mpfr.js (the latter is currently a trivial 10 liner), things might improve...

## Python

Slowed to a crawl at n of 1267909, so did not run for larger n.

`from math import isqrt def juggler(k, countdig=True, maxiters=1000):    m, maxj, maxjpos = k, k, 0    for i in range(1, maxiters):        m = isqrt(m) if m % 2 == 0 else isqrt(m * m * m)        if m >= maxj:            maxj, maxjpos  = m, i        if m == 1:            print(f"{k: 9}{i: 6,}{maxjpos: 6}{len(str(maxj)) if countdig else maxj: 20,}{' digits' if countdig else ''}")            return i     print("ERROR: Juggler series starting with \$k did not converge in \$maxiters iterations")  print("       n    l(n)  i(n)       h(n) or d(n)\n-------------------------------------------")for k in range(20, 40):    juggler(k, False) for k in [113, 173, 193, 2183, 11229, 15065, 15845, 30817, 48443, 275485, 1267909]:    juggler(k) `
Output:
```       n    l(n)  i(n)         h(n) or d(n)
--------------------------------------------
20     3     0                  20
21     9     4                 140
22     3     0                  22
23     9     1                 110
24     3     0                  24
25    11     3              52,214
26     6     3                  36
27     6     1                 140
28     6     3                  36
29     9     1                 156
30     6     3                  36
31     6     1                 172
32     6     3                  36
33     8     2               2,598
34     6     3                  36
35     8     2               2,978
36     3     0                  36
37    17     8  24,906,114,455,136
38     3     0                  38
39    14     3             233,046
113    16     9                  27 digits
173    32    17                  82 digits
193    73    47                 271 digits
2183    72    32               5,929 digits
11229   101    54               8,201 digits
15065    66    25              11,723 digits
15845   139    43              23,889 digits
30817    93    39              45,391 digits
48443   157    60             972,463 digits
275485   225   148           1,909,410 digits
1267909   151    99           1,952,329 digits
```

## Quackery

`  [ dip number\$     over size -     space swap of    swap join echo\$ ]          is recho   ( n n -->   )   [ 1 & ]                      is odd     (   n --> b )   [ dup dup * * ]              is cubed   (   n --> n )   [ dup 1    [ 2dup > while      + 1 >>      2dup / again ]    drop nip ]                 is sqrt    (   n --> n )   [ nested     [ dup -1 peek 1 != while      dup -1 peek       dup odd if cubed      sqrt join again ] ]      is juggler (   n --> [ )    [ dup 4 recho    juggler    dup size 1 -     3 recho    0 swap behead swap     witheach       [ 2dup < if          [ rot drop             i^ 1+ unrot            swap ]        drop ]      15 recho 2 recho cr ]     is stats   (   n -->   )   20 times [ i^ 20 + stats ]`
Output:
```  20  3             20 0
21  9            140 4
22  3             22 0
23  9            110 1
24  3             24 0
25 11          52214 3
26  6             36 3
27  6            140 1
28  6             36 3
29  9            156 1
30  6             36 3
31  6            172 1
32  6             36 3
33  8           2598 2
34  6             36 3
35  8           2978 2
36  3             36 0
37 17 24906114455136 8
38  3             38 0
39 14         233046 3```

## Raku

Reaches 30817 fairly quickly but later values suck up enough memory that it starts thrashing the disk cache and performance drops off a cliff (on my system). Killed it after 10 minutes and capped list at 30817. Could rewrite to not try to hold entire sequence in memory at once, but probably not worth it. If you want sheer numeric calculation performance, Raku is probably not where it's at.

`use Lingua::EN::Numbers;sub juggler (Int \$n where * > 0) { \$n, { \$_ +& 1 ?? .³.&isqrt !! .&isqrt } … 1 } sub isqrt ( \x ) { my ( \$X, \$q, \$r, \$t ) =  x, 1, 0 ;    \$q +<= 2 while \$q ≤ \$X ;    while \$q > 1 {        \$q +>= 2; \$t = \$X - \$r - \$q; \$r +>= 1;        if \$t ≥ 0 { \$X = \$t; \$r += \$q }    }    \$r} say " n  l[n]  i[n]   h[n]";for 20..39 {    my @j = .&juggler;    my \$max = @j.max;    printf "%2s %4d  %4d    %s\n", .&comma, +@j-1, @j.first(* == \$max, :k), comma \$max;} say "\n      n     l[n]   i[n]    d[n]";( 113, 173, 193, 2183, 11229, 15065, 15845, 30817 ).hyper(:1batch).map: {    my \$start = now;    my @j = .&juggler;    my \$max = @j.max;    printf "%10s %4d   %4d %10s   %6.2f seconds\n", .&comma, +@j-1, @j.first(* == \$max, :k),      \$max.chars.&comma, (now - \$start);}`
Output:
``` n  l[n]  i[n]   h[n]
20    3     0    20
21    9     4    140
22    3     0    22
23    9     1    110
24    3     0    24
25   11     3    52,214
26    6     3    36
27    6     1    140
28    6     3    36
29    9     1    156
30    6     3    36
31    6     1    172
32    6     3    36
33    8     2    2,598
34    6     3    36
35    8     2    2,978
36    3     0    36
37   17     8    24,906,114,455,136
38    3     0    38
39   14     3    233,046

n     l[n]   i[n]    d[n]
113   16      9         27     0.01 seconds
173   32     17         82     0.01 seconds
193   73     47        271     0.09 seconds
2,183   72     32      5,929     1.05 seconds
11,229  101     54      8,201     1.98 seconds
15,065   66     25     11,723     2.05 seconds
15,845  139     43     23,889    10.75 seconds
30,817   93     39     45,391    19.60 seconds```

## REXX

REXX doesn't have a native integer sqrt function, so one was utilized that was previously written.

Also, one optimization was to examine the last decimal digit for   oddness   (instead of getting the remainder when
dividing by two).

Another optimization was to reduce the number of digits after the   sqrt   was calculated.

`/*REXX program  calculates and displays  the  juggler sequence  for any positive integer*/numeric digits 20                                /*define the number of decimal digits. */parse arg LO HI list                             /*obtain optional arguments from the CL*/if LO='' | LO=","  then do; LO= 20; HI= 39; end  /*Not specified?  Then use the default.*/if HI='' | HI=","  then HI= LO                   /* "      "         "   "   "     "    */w= length( commas(HI) )                          /*find the max width of any number  N. */d= digits();     dd= d + d%3 + 1                 /*get # numeric digits; another version*/if LO>0  then say c('n',w)    c("steps",7)    c('maxIndex',8)   c("biggest term"    ,dd)if LO>0  then say c('' ,w,.)  c(""     ,7,.)  c(''        ,8,.) c(""                ,dd,.)     do n=LO  to HI while n>0; steps= juggler(n)  /*invoke JUGGLER: find the juggler num.*/    nc= commas(n)                                /*maybe add commas to  N.              */    say right(nc, w)      c(steps, 8)     c(imx, 8)      right( commas(mx), dd-1)    end   /*n*/                                                 /*biggest term isn't shown for list #s.*/xtra= words(list)                                /*determine how many numbers in list.  */if xtra==0  then exit 0                          /*no special ginormous juggler numbers?*/w= max(9, length( commas( word(list, xtra) ) ) ) /*find the max width of any list number*/say;             say;             saysay c('n',w)     c("steps",7)     c('maxIndex',8)     c("max number of digits",dd)say c('' ,w,.)   c(""     ,7,.)   c(''        ,8,.)   c(""                    ,dd,.)          do p=1  for xtra;     n= word(list, p)  /*process each of the numbers in list. */         steps= juggler(n);    nc= commas(n)     /*get a number & pass it on to JUGGLER.*/         say right(nc, w)   c(steps, 8)   c(imx, 8)      right( commas(length(mx)), dd-1)         end   /*p*/exit 0                                           /*stick a fork in it,  we're all done. *//*──────────────────────────────────────────────────────────────────────────────────────*/c:      parse arg c1,c2,c3; if c3==''  then c3= ' '; else c3= '─'; return center(c1,c2,c3)commas: parse arg ?;  do jc=length(?)-3  to 1  by -3; ?=insert(',', ?, jc); end;  return ?/*──────────────────────────────────────────────────────────────────────────────────────*/Isqrt:  procedure; parse arg x;  q= 1;  r= 0     /*obtain X;  R  will be the Isqrt(X).  */           do until q>x;  q= q * 4               /*multiply   Q   by four until > X.    */           end   /*until*/               do while q>1;  q= q % 4           /*processing while Q>1;  quarter Q.    */               t= x - r - q;  r= r % 2           /*set T ──► X-R-Q;       halve R.      */               if t>=0  then do; x= t; r= r + q  /*T≥0?  Then set X ──► T;  add Q ──► R.*/                             end               end   /*while*/;    return r      /*return the integer square root of  X.*//*──────────────────────────────────────────────────────────────────────────────────────*/juggler: parse arg #;  imx= 0;  mx= #            /*get N;  set index of MX and MX ──► 0.*/         @.=0; @.1=1; @.3=1; @.5=1; @.7=1; @.9=1 /*set semaphores (≡1) for odd dec. digs*/           do j=1  until z==1                    /*here is where we begin to juggle.    */           parse var  #  ''  -1  _               /*obtain the last decimal digit of  #. */           if @._  then do;       cube= #*#*#    /*Odd? Then compute integer sqrt(#**3).*/                        if pos(., cube)>0  then do    /*need to increase decimal digits.*/                                                parse var  cube    with    'E'  x                                                if x>=digits()  then numeric digits x+2                                                end                        z= Isqrt(#*#*#)          /*compute the integer sqrt(#**3)       */                        end                   else z= Isqrt(#)              /*Even? Then compute integer sqrt(#).  */           L= length(z)           if L>=d  then numeric digits L+1      /*reduce the number of numeric digits. */           if z>mx  then do;  mx= z; imx= j; end /*found a new max;  set MX;  set IMX.  */           #= z           end   /*j*/;                 return j`
output   when using the inputs:     ,   ,   113   173   193   2183   11229   15065   30817   48443
```n   steps  maxIndex        biggest term
── ─────── ──────── ───────────────────────────
20    3        0                             20
21    9        4                            140
22    3        0                             22
23    9        1                            110
24    3        0                             24
25    11       3                         52,214
26    6        3                             36
27    6        1                            140
28    6        3                             36
29    9        1                            156
30    6        3                             36
31    6        1                            172
32    6        3                             36
33    8        2                          2,598
34    6        3                             36
35    8        2                          2,978
36    3        0                             36
37    17       8             24,906,114,455,136
38    3        0                             38
39    14       3                        233,046

n      steps  maxIndex    max number of digits
───────── ─────── ──────── ───────────────────────────
113    16       9                             27
173    32       17                            82
193    73       47                           271
2,183    72       32                         5,929
11,229   101       54                         8,201
15,065    66       25                        11,723
15,845   139       43                        23,889
30,817    93       39                        45,391
48,443   157       60                       972,463
```

## Ruby

`def juggler(k) = k.even? ? Integer.sqrt(k) : Integer.sqrt(k*k*k) (20..39).chain([113, 173, 193, 2183, 11229, 15065, 15845, 30817, 48443, 275485, 1267909, 2264915]).each do |k|  k1 = k  l = h = i = 0  until k == 1 do    h, i = k, l if k > h    l += 1    k =  juggler(k)  end  if k1 < 40 then    puts "#{k1}: l[n] = #{l}, h[n] = #{h}, i[n] = #{i}"  else    puts "#{k1}: l[n] = #{l}, d[n] = #{h.to_s.size}, i[n] = #{i}"  endend `
Output:
```20: l[n] = 3, h[n] = 20, i[n] = 0
21: l[n] = 9, h[n] = 140, i[n] = 4
22: l[n] = 3, h[n] = 22, i[n] = 0
23: l[n] = 9, h[n] = 110, i[n] = 1
24: l[n] = 3, h[n] = 24, i[n] = 0
25: l[n] = 11, h[n] = 52214, i[n] = 3
26: l[n] = 6, h[n] = 36, i[n] = 3
27: l[n] = 6, h[n] = 140, i[n] = 1
28: l[n] = 6, h[n] = 36, i[n] = 3
29: l[n] = 9, h[n] = 156, i[n] = 1
30: l[n] = 6, h[n] = 36, i[n] = 3
31: l[n] = 6, h[n] = 172, i[n] = 1
32: l[n] = 6, h[n] = 36, i[n] = 3
33: l[n] = 8, h[n] = 2598, i[n] = 2
34: l[n] = 6, h[n] = 36, i[n] = 3
35: l[n] = 8, h[n] = 2978, i[n] = 2
36: l[n] = 3, h[n] = 36, i[n] = 0
37: l[n] = 17, h[n] = 24906114455136, i[n] = 8
38: l[n] = 3, h[n] = 38, i[n] = 0
39: l[n] = 14, h[n] = 233046, i[n] = 3
113: l[n] = 16, d[n] = 27, i[n] = 9
173: l[n] = 32, d[n] = 82, i[n] = 17
193: l[n] = 73, d[n] = 271, i[n] = 47
2183: l[n] = 72, d[n] = 5929, i[n] = 32
11229: l[n] = 101, d[n] = 8201, i[n] = 54
15065: l[n] = 66, d[n] = 11723, i[n] = 25
15845: l[n] = 139, d[n] = 23889, i[n] = 43
30817: l[n] = 93, d[n] = 45391, i[n] = 39
48443: l[n] = 157, d[n] = 972463, i[n] = 60
275485: l[n] = 225, d[n] = 1909410, i[n] = 148
1267909: l[n] = 151, d[n] = 1952329, i[n] = 99
2264915: l[n] = 149, d[n] = 2855584, i[n] = 89
```

## Wren

### Wren CLI

Library: Wren-fmt
Library: Wren-big

This took just over 17 minutes to reach n = 30,817 on my machine and I gave up after that.

`import "/fmt" for Fmtimport "/big" for BigInt var one  = BigInt.one var juggler = Fn.new { |n|    if (n < 1) Fiber.abort("Starting value must be a positive integer.")    var a = BigInt.new(n)    var count = 0    var maxCount = 0    var max = a.copy()    while (a != one) {        if (a.isEven) {            a = a.isqrt        } else {            a = (a.square * a).isqrt        }        count = count + 1        if (a > max) {            max = a            maxCount = count        }    }    return [count, maxCount, max, max.toString.count]} System.print("n    l[n]  i[n]  h[n]")System.print("-----------------------------------") for (n in 20..39) {    var res = juggler.call(n)    Fmt.print("\$2d    \$2d   \$2d    \$,i", n, res[0], res[1], res[2])}System.print()var nums = [113, 173, 193, 2183, 11229, 15065, 15845, 30817]System.print("   n     l[n]   i[n]   d[n]")System.print("----------------------------")for (n in nums) {    var res = juggler.call(n)    Fmt.print("\$,6d   \$3d    \$3d   \$,6i", n, res[0], res[1], res[3])}`
Output:
```n    l[n]  i[n]  h[n]
-----------------------------------
20     3    0    20
21     9    4    140
22     3    0    22
23     9    1    110
24     3    0    24
25    11    3    52,214
26     6    3    36
27     6    1    140
28     6    3    36
29     9    1    156
30     6    3    36
31     6    1    172
32     6    3    36
33     8    2    2,598
34     6    3    36
35     8    2    2,978
36     3    0    36
37    17    8    24,906,114,455,136
38     3    0    38
39    14    3    233,046

n        l[n]   i[n]   d[n]
----------------------------
113    16      9       27
173    32     17       82
193    73     47      271
2,183    72     32    5,929
11,229   101     54    8,201
15,065    66     25   11,723
15,845   139     43   23,889
30,817    93     39   45,391
```

### Embedded

Library: Wren-gmp

Massive speed-up, of course, when one brings in GMP. Now takes about 1 minute 48 seconds to reach 7,110,201 which is not much slower than Go on the same machine!

`/* juggler-gmp.wren */ import "./gmp" for Mpzimport "./fmt" for Fmt var one  = Mpz.one var juggler = Fn.new { |n|    if (n < 1) Fiber.abort("Starting value must be a positive integer.")    var a = Mpz.from(n)    var count = 0    var maxCount = 0    var max = a.copy()    while (a != one) {        if (a.isEven) {            a.sqrt        } else {            a.cube.sqrt        }        count = count + 1        if (a > max) {            max.set(a)            maxCount = count        }    }    return [count, maxCount, max, max.toString.count]} System.print("n    l[n]  i[n]  h[n]")System.print("-----------------------------------")for (n in 20..39) {    var res = juggler.call(n)    Fmt.print("\$2d    \$2d   \$2d    \$,i", n, res[0], res[1], res[2])}System.print()var nums = [    113, 173, 193, 2183, 11229, 15065, 15845, 30817, 48443,    275485, 1267909, 2264915, 5812827, 7110201] System.print("     n      l[n]   i[n]   d[n]")System.print("-----------------------------------")for (n in nums) {    var res = juggler.call(n)    Fmt.print("\$,9d   \$3d    \$3d    \$,i", n, res[0], res[1], res[3])}`
Output:
```n    l[n]  i[n]  h[n]
-----------------------------------
20     3    0    20
21     9    4    140
22     3    0    22
23     9    1    110
24     3    0    24
25    11    3    52,214
26     6    3    36
27     6    1    140
28     6    3    36
29     9    1    156
30     6    3    36
31     6    1    172
32     6    3    36
33     8    2    2,598
34     6    3    36
35     8    2    2,978
36     3    0    36
37    17    8    24,906,114,455,136
38     3    0    38
39    14    3    233,046

n      l[n]   i[n]   d[n]
-----------------------------------
113    16      9    27
173    32     17    82
193    73     47    271
2,183    72     32    5,929
11,229   101     54    8,201
15,065    66     25    11,723
15,845   139     43    23,889
30,817    93     39    45,391
48,443   157     60    972,463
275,485   225    148    1,909,410
1,267,909   151     99    1,952,329
2,264,915   149     89    2,855,584
5,812,827   135     67    7,996,276
7,110,201   205    119    89,981,517
```