Smallest power of 6 whose decimal expansion contains n

From Rosetta Code
Smallest power of 6 whose decimal expansion contains n is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.
Task

Show the smallest (non-negative integer) power of   6   whose decimal expansion contains   n,     where   n   <   22


11l[edit]

Translation of: Python
F smallest_six(n)
   V p = BigInt(1)
   L String(n) !C String(p)
      p *= 6
   R p

L(n) 22
   print(‘#2: #.’.format(n, smallest_six(n)))
Output:
 0: 10077696
 1: 1
 2: 216
 3: 36
 4: 46656
 5: 46656
 6: 6
 7: 7776
 8: 2176782336
 9: 1296
10: 10077696
11: 2821109907456
12: 1296
13: 13060694016
14: 6140942214464815497216
15: 101559956668416
16: 216
17: 60466176
18: 470184984576
19: 21936950640377856
20: 170581728179578208256
21: 216

ALGOL 68[edit]

Works with: ALGOL 68G version Any - tested with release 2.8.3.win32

Uses ALGOL 68G's LONG LONG INT large integers, the default precision is sufficient for this task. Also uses the ALGOL 68G specific string in string procedure.

BEGIN # find the smallest k such that the decimal representation of 6^k contains n for 0 <= n <= 21 #
    # returns s blank-padded on the right to at least len characters #
    PROC right pad = ( STRING s, INT len )STRING:
         BEGIN
            INT s len = ( UPB s - LWB s ) + 1;
            IF s len >= len THEN s ELSE s + ( len - s len ) * " " FI
         END # right pad # ;
    # returns s blank-padded on the left to at least len characters #
    PROC left pad = ( STRING s, INT len )STRING:
         BEGIN
            INT s len = ( UPB s - LWB s ) + 1;
            IF s len >= len THEN s ELSE ( ( len - s len ) * " " ) + s FI
         END # left pad # ;
    # returns a string representation of unformatted with space separators                    #
    PROC space separate = ( STRING unformatted )STRING:
         BEGIN
            STRING result      := "";
            INT    ch count    := 0;
            FOR c FROM UPB unformatted BY -1 TO LWB unformatted DO
                IF   ch count <= 2 THEN ch count +:= 1
                ELSE                    ch count  := 1; " " +=: result
                FI;
                unformatted[ c ] +=: result
            OD;
            result
         END # space separate # ;
    # start with powers up to 6^12, if this proves insufficient, the kk array will be extended #
    FLEX[ 0 : 12 ]STRING kk;
    FOR k FROM LWB kk TO UPB kk DO kk[ k ] := whole( LONG LONG INT( 6 ) ^ k, 0 ) OD;
    # find the numbers #
    FOR i FROM 0 TO 21 DO
        STRING n      = whole( i, 0 );
        BOOL try again := TRUE;
        WHILE try again DO
            try again := FALSE;
            BOOL   found := FALSE;
            FOR k FROM LWB kk TO UPB kk WHILE NOT found DO
                IF string in string( n, NIL, kk[ k ] ) THEN
                    found := TRUE;
                    print( ( whole( i, -2 ), right pad( ": 6^" + whole( k, 0 ), 8 ), " ", left pad( space separate( kk[ k ] ), 30 ), newline ) )
                FI
            OD;
            IF NOT found THEN
                # haven't got enough k^k values - get some more #
                kk := HEAP[ 1 : UPB kk * 2 ]STRING;
                FOR k FROM LWB kk TO UPB kk DO kk[ k ] := whole( LONG LONG INT( 6 ) ^ k, 0 ) OD;
                try again := TRUE
            FI
        OD
    OD
END
Output:
 0: 6^9                        10 077 696
 1: 6^0                                 1
 2: 6^3                               216
 3: 6^2                                36
 4: 6^6                            46 656
 5: 6^6                            46 656
 6: 6^1                                 6
 7: 6^5                             7 776
 8: 6^12                    2 176 782 336
 9: 6^4                             1 296
10: 6^9                        10 077 696
11: 6^16                2 821 109 907 456
12: 6^4                             1 296
13: 6^13                   13 060 694 016
14: 6^28    6 140 942 214 464 815 497 216
15: 6^18              101 559 956 668 416
16: 6^3                               216
17: 6^10                       60 466 176
18: 6^15                  470 184 984 576
19: 6^21           21 936 950 640 377 856
20: 6^26      170 581 728 179 578 208 256
21: 6^3                               216

ALGOL W[edit]

Algol W doesn't have integers larger than 32 bits, however we can handle the required numbers with arrays of digits.

begin % find the smallest power of 6 that contains n for 0 <= n <= 21       %
    % we assume that powers of 6 upto 6^32 will be sufficient               %
    % as Algol W does not have integers longer than 32 bits, the powers     %
    % will be held in an array where each element is a single digit of the  %
    % power, the least significant digit of 6^n is in powers( n, 1 )        %
    integer array powers ( 0 :: 32, 1 :: 32 ); % the powers                 %
    integer array digits ( 0 :: 32 ); % the number of digits in each power  %
    integer array lowest ( 0 :: 21 ); % the lowest power containing the idx %
    for n := 0 until 21 do lowest( n ) := -1;
    % 6^0 = 1, which is the lowest power containing 1                       %
    lowest( 1 )    := 0;
    powers( 0, 1 ) := 1;
    for d := 2 until 32 do powers( 0, d ) := 0;
    digits( 0 ) := 1;
    % calculate the remaining powers and find the numbers 0..21             %
    for p := 1 until 32 do begin
        integer carry, dPos, dMax;
        dPos  := 1;
        dMax  := digits( p - 1 );
        carry := 0;
        % compute the power p and find the single digit numbers             %
        while dPos <= dMax do begin
            integer d;
            d                 := carry + ( powers( p - 1, dPos ) * 6 );
            carry             := d div 10;
            d                 := d rem 10;
            if lowest( d ) < 0 then lowest( d ) := p;
            powers( p, dPos ) := d;
            dPos              := dPos + 1
        end while_dPos_le_dMax ;
        if   carry = 0
        then digits( p ) := dMax
        else begin
            % the power p has one more digit than the previous              %
            digits( p )       := dPos;
            powers( p, dPos ) := carry;
            if lowest( carry ) < 0 then lowest( carry ) := p;
        end if_carry_eq_0__ ;
        % find the two digit numbers                                        %
        for n := 10 until 21 do begin
            if lowest( n ) < 0 then begin
                integer h, l;
                h := n div 10;
                l := n rem 10;
                for d := digits( p ) - 1 step -1 until 1 do begin
                    if powers( p, d ) = l and powers( p, d + 1 ) = h then lowest( n ) := p
                end for_d
            end if_lowest_n_lt_0
        end for_n
    end for_p ;
    % show the lowest powers that contain the numbers 0..21                  %
    for n := 0 until 21 do begin
        integer p;
        p := lowest( n );
        write( i_w := 2, s_w := 0, n, " in 6^", p, ": " );
        for d := digits( p ) step -1 until 1 do writeon( i_w := 1, s_w := 0, powers( p, d ) )
    end for_n
end.
Output:
 0 in 6^ 9: 10077696
 1 in 6^ 0: 1
 2 in 6^ 3: 216
 3 in 6^ 2: 36
 4 in 6^ 6: 46656
 5 in 6^ 6: 46656
 6 in 6^ 1: 6
 7 in 6^ 5: 7776
 8 in 6^12: 2176782336
 9 in 6^ 4: 1296
10 in 6^ 9: 10077696
11 in 6^16: 2821109907456
12 in 6^ 4: 1296
13 in 6^13: 13060694016
14 in 6^28: 6140942214464815497216
15 in 6^18: 101559956668416
16 in 6^ 3: 216
17 in 6^10: 60466176
18 in 6^15: 470184984576
19 in 6^21: 21936950640377856
20 in 6^26: 170581728179578208256
21 in 6^ 3: 216

AWK[edit]

# syntax: GAWK -f SMALLEST_POWER_OF_6_WHOSE_DECIMAL_EXPANSION_CONTAINS_N.AWK
BEGIN {
    printf(" n power %30s\n","smallest power of 6")
    for (n=0; n<22; n++) {
      p = 1
      power = 0
      while (p !~ n) {
        p *= 6
        power++
      }
      printf("%2d %5d %'30d\n",n,power,p)
    }
    exit(0)
}
Output:
 n power            smallest power of 6
 0     9                     10,077,696
 1     0                              1
 2     3                            216
 3     2                             36
 4     6                         46,656
 5     6                         46,656
 6     1                              6
 7     5                          7,776
 8    12                  2,176,782,336
 9     4                          1,296
10     9                     10,077,696
11    16              2,821,109,907,456
12     4                          1,296
13    13                 13,060,694,016
14    28  6,140,942,214,464,815,497,216
15    18            101,559,956,668,416
16     3                            216
17    10                     60,466,176
18    15                470,184,984,576
19    21         21,936,950,640,377,856
20    26    170,581,728,179,578,208,256
21     3                            216

C[edit]

#include <stdio.h>
#include <string.h>
#include <gmp.h>

char *power_of_six(unsigned int n, char *buf) {
    mpz_t p;
    mpz_init(p);
    mpz_ui_pow_ui(p, 6, n);
    mpz_get_str(buf, 10, p);
    mpz_clear(p);
    return buf;
}

char *smallest_six(unsigned int n) {
    static char nbuf[32], powbuf[1024];
    unsigned int p = 0;
    
    do {
        sprintf(nbuf, "%u", n);
        power_of_six(p++, powbuf);
    } while (!strstr(powbuf, nbuf));
    
    return powbuf;
}

int main() {
    unsigned int i;
    
    for (i=0; i<22; i++) {
        printf("%d: %s\n", i, smallest_six(i));
    }
    
    return 0;
}
Output:
0: 10077696
1: 1
2: 216
3: 36
4: 46656
5: 46656
6: 6
7: 7776
8: 2176782336
9: 1296
10: 10077696
11: 2821109907456
12: 1296
13: 13060694016
14: 6140942214464815497216
15: 101559956668416
16: 216
17: 60466176
18: 470184984576
19: 21936950640377856
20: 170581728179578208256
21: 216

C++[edit]

#include <iostream>
#include <iomanip>
#include <string>
#include <gmpxx.h>

std::string smallest_six(unsigned int n) {
    mpz_class pow = 1;
    std::string goal = std::to_string(n);
    
    while (pow.get_str().find(goal) == std::string::npos) {
        pow *= 6;
    }
    
    return pow.get_str();
}

int main() {
    for (unsigned int i=0; i<22; i++) {
        std::cout << std::setw(2) << i << ": " 
                  << smallest_six(i) << std::endl;
    }
    return 0;
}
Output:
 0: 10077696
 1: 1
 2: 216
 3: 36
 4: 46656
 5: 46656
 6: 6
 7: 7776
 8: 2176782336
 9: 1296
10: 10077696
11: 2821109907456
12: 1296
13: 13060694016
14: 6140942214464815497216
15: 101559956668416
16: 216
17: 60466176
18: 470184984576
19: 21936950640377856
20: 170581728179578208256
21: 216

CLU[edit]

% This program uses the bigint type that comes with PCLU.
% It is in "misc.lib"
%
% pclu -merge $CLUHOME/lib/misc.lib -compile n6_contains_6.clu

smallest_power_6 = proc (n: int) returns (int, bigint)
    n_str: string := int$unparse(n)
    six_power: bigint := bigint$i2bi(1)
    six: bigint := bigint$i2bi(6)
    n_power: int := 0
    
    while true do
        pow_str: string := bigint$unparse(six_power)
        if string$indexs(n_str, pow_str) ~= 0 then 
            return(n_power, six_power)
        end
        six_power := six_power * six
        n_power := n_power + 1
    end
end smallest_power_6

start_up = proc ()
    po: stream := stream$primary_output()
    
    for n: int in int$from_to(0, 21) do
        p: int  val: bigint
        stream$putright(po, int$unparse(n), 2)
        stream$puts(po, ": 6^")
        p, val := smallest_power_6(n)
        stream$putleft(po, int$unparse(p), 2)
        stream$puts(po, " = ")
        stream$putright(po, bigint$unparse(val), 30)
        stream$putl(po, "")
    end
end start_up
Output:
 0: 6^9  =                       10077696
 1: 6^0  =                              1
 2: 6^3  =                            216
 3: 6^2  =                             36
 4: 6^6  =                          46656
 5: 6^6  =                          46656
 6: 6^1  =                              6
 7: 6^5  =                           7776
 8: 6^12 =                     2176782336
 9: 6^4  =                           1296
10: 6^9  =                       10077696
11: 6^16 =                  2821109907456
12: 6^4  =                           1296
13: 6^13 =                    13060694016
14: 6^28 =         6140942214464815497216
15: 6^18 =                101559956668416
16: 6^3  =                            216
17: 6^10 =                       60466176
18: 6^15 =                   470184984576
19: 6^21 =              21936950640377856
20: 6^26 =          170581728179578208256
21: 6^3  =                            216

F#[edit]

// Nigel Galloway. April 9th., 2021
let rec fN i g e l=match l%i=g,l/10I with (true,_)->e |(_,l) when l=0I->fN i g (e*6I) (e*6I) |(_,l)->fN i g e l 
[0I..99I]|>Seq.iter(fun n->printfn "%2d %A" (int n)(fN(if n>9I then 100I else 10I) n 1I 1I))
Output:
 0 10077696
 1 1
 2 216
 3 36
 4 46656
 5 46656
 6 6
 7 7776
 8 2176782336
 9 1296
10 10077696
11 2821109907456
12 1296
13 13060694016
14 6140942214464815497216
15 101559956668416
16 216
17 60466176
18 470184984576
19 21936950640377856
20 170581728179578208256
21 216
22 131621703842267136
23 2176782336
24 1023490369077469249536
25 170581728179578208256
26 16926659444736
27 279936
28 2821109907456
29 1296
30 13060694016
31 131621703842267136
32 4738381338321616896
33 2176782336
34 1023490369077469249536
35 609359740010496
36 36
37 21936950640377856
38 131621703842267136
39 221073919720733357899776
40 13060694016
41 78364164096
42 131621703842267136
43 28430288029929701376
44 16926659444736
45 470184984576
46 46656
47 470184984576
48 6140942214464815497216
49 470184984576
50 21936950640377856
51 1326443518324400147398656
52 623673825204293256669089197883129856
53 789730223053602816
54 6140942214464815497216
55 101559956668416
56 46656
57 470184984576
58 3656158440062976
59 16926659444736
60 60466176
61 1679616
62 362797056
63 47751966659678405306351616
64 78364164096
65 46656
66 46656
67 1679616
68 101559956668416
69 10077696
70 362797056
71 131621703842267136
72 170581728179578208256
73 16926659444736
74 2821109907456
75 47751966659678405306351616
76 7776
77 7776
78 2176782336
79 279936
80 28430288029929701376
81 789730223053602816
82 2176782336
83 78364164096
84 470184984576
85 21936950640377856
86 36845653286788892983296
87 61886548790943213277031694336
88 28430288029929701376
89 789730223053602816
90 2821109907456
91 221073919720733357899776
92 16926659444736
93 279936
94 13060694016
95 101559956668416
96 1296
97 362797056
98 470184984576
99 279936
Real: 00:00:00.066

Factor[edit]

Works with: Factor version 0.99 2021-02-05
USING: formatting kernel lists lists.lazy math math.functions
present sequences tools.memory.private ;

: powers-of-6 ( -- list )
    0 lfrom [ 6 swap ^ ] lmap-lazy ;

: smallest ( m -- n )
    present powers-of-6 [ present subseq? ] with lfilter car ;

22 [ dup smallest commas "%2d   %s\n" printf ] each-integer
Output:
 0   10,077,696
 1   1
 2   216
 3   36
 4   46,656
 5   46,656
 6   6
 7   7,776
 8   2,176,782,336
 9   1,296
10   10,077,696
11   2,821,109,907,456
12   1,296
13   13,060,694,016
14   6,140,942,214,464,815,497,216
15   101,559,956,668,416
16   216
17   60,466,176
18   470,184,984,576
19   21,936,950,640,377,856
20   170,581,728,179,578,208,256
21   216


FreeBASIC[edit]

Translation of: Ring
Print !"\ntrabajando...\n"
Print !"M¡nima potencia de 6 cuya expansi¢n decimal contiene n:\n"

Dim As Uinteger num = 0, limit = 200, m

For n As Ubyte = 0 To 21
    Dim As String strn = Str(n)
    For m = 0 To limit
        Dim As String strpow = Str(6 ^ m)
        Dim As Ulong ind = Instr(strpow,strn)
        If ind > 0 Then
            Print Using "##. 6^\\ = &"; n; Str(m); strpow
            Exit For
        End If
    Next m
Next n

Print !"\n--- terminado, pulsa RETURN---"
Sleep


Go[edit]

Translation of: Wren
package main

import (
    "fmt"
    "math/big"
    "strconv"
    "strings"
)

// Adds thousand separators to an integral string.
func commatize(s string) string {
    neg := false
    if strings.HasPrefix(s, "-") {
        s = s[1:]
        neg = true
    }
    le := len(s)
    for i := le - 3; i >= 1; i -= 3 {
        s = s[0:i] + "," + s[i:]
    }
    if !neg {
        return s
    }
    return "-" + s
}

func main() {
    fmt.Println(" n  smallest power of 6 which contains n")
    six := big.NewInt(6)
    for n := 0; n <= 21; n++ {
        ns := strconv.Itoa(n)
        i := int64(0)
        for {
            bi := big.NewInt(i)
            pow6 := bi.Exp(six, bi, nil).String()
            if strings.Contains(pow6, ns) {
                fmt.Printf("%2d  6^%-2d = %s\n", n, i, commatize(pow6))
                break
            }
            i++
        }
    }
}
Output:
 n  smallest power of 6 which contains n
 0  6^9  = 10,077,696
 1  6^0  = 1
 2  6^3  = 216
 3  6^2  = 36
 4  6^6  = 46,656
 5  6^6  = 46,656
 6  6^1  = 6
 7  6^5  = 7,776
 8  6^12 = 2,176,782,336
 9  6^4  = 1,296
10  6^9  = 10,077,696
11  6^16 = 2,821,109,907,456
12  6^4  = 1,296
13  6^13 = 13,060,694,016
14  6^28 = 6,140,942,214,464,815,497,216
15  6^18 = 101,559,956,668,416
16  6^3  = 216
17  6^10 = 60,466,176
18  6^15 = 470,184,984,576
19  6^21 = 21,936,950,640,377,856
20  6^26 = 170,581,728,179,578,208,256
21  6^3  = 216

Haskell[edit]

import Data.List (isInfixOf)
import Text.Printf (printf)

sixes :: [Integer]
sixes = iterate (* 6) 1

smallest :: Integer -> Integer
smallest n =
  head $
    filter
      ((show n `isInfixOf`) . show)
      sixes

main :: IO ()
main =
  putStr $
    concatMap
      (printf "%2d: %d\n" <*> smallest)
      [0 .. 21]
Output:
 0: 10077696
 1: 1
 2: 216
 3: 36
 4: 46656
 5: 46656
 6: 6
 7: 7776
 8: 2176782336
 9: 1296
10: 10077696
11: 2821109907456
12: 1296
13: 13060694016
14: 6140942214464815497216
15: 101559956668416
16: 216
17: 60466176
18: 470184984576
19: 21936950640377856
20: 170581728179578208256
21: 216

jq[edit]

Works with gojq, the Go implementation of jq

gojq provides unbounded-precision integer arithmetic and is therefore appropriate for this task.

# To preserve precision:
def power($b): . as $in | reduce range(0;$b) as $i (1; . * $in);

range(0;22)
| . as $in
| tostring as $n 
| first(range(0;infinite) as $i | 6 | power($i) | . as $p | tostring | (index($n) // empty) 
        | [$in,$i,$p] )
Output:
[0,9,10077696]
[1,0,1]
[2,3,216]
[3,2,36]
[4,6,46656]
[5,6,46656]
[6,1,6]
[7,5,7776]
[8,12,2176782336]
[9,4,1296]
[10,9,10077696]
[11,16,2821109907456]
[12,4,1296]
[13,13,13060694016]
[14,28,6140942214464815497216]
[15,18,101559956668416]
[16,3,216]
[17,10,60466176]
[18,15,470184984576]
[19,21,21936950640377856]
[20,26,170581728179578208256]
[21,3,216]


Julia[edit]

using Formatting

digcontains(n, dig) = contains(String(Char.(digits(n))), String(Char.(dig)))

function findpow6containing(needle)
    dig = digits(needle)
    for i in 0:1000
        p = big"6"^i
        digcontains(p, dig) && return p
    end
    error("could not find a  power of 6 containing $dig")
end

for n in 0:21
    println(rpad(n, 5), format(findpow6containing(n), commas=true))
end
Output:
0    10,077,696
1    1
2    216
3    36
4    46,656
5    46,656
6    6
7    7,776
8    2,176,782,336
9    1,296
10   10,077,696
11   2,821,109,907,456
12   1,296
13   13,060,694,016
14   6,140,942,214,464,815,497,216
15   101,559,956,668,416
16   216
17   60,466,176
18   470,184,984,576
19   21,936,950,640,377,856
20   170,581,728,179,578,208,256
21   216

Mathematica/Wolfram Language[edit]

ClearAll[SmallestPowerContainingN]
SmallestPowerContainingN[n_Integer] := Module[{i = 1, test},
  While[True,
   test = 6^i;
   If[SequenceCount[IntegerDigits[test], IntegerDigits[n]] > 0, 
    Return[{n, i, test}]];
   i++;
   ]
  ]
Grid[SmallestPowerContainingN /@ Range[0, 21]]
Output:
0	9	10077696
1	3	216
2	3	216
3	2	36
4	6	46656
5	6	46656
6	1	6
7	5	7776
8	12	2176782336
9	4	1296
10	9	10077696
11	16	2821109907456
12	4	1296
13	13	13060694016
14	28	6140942214464815497216
15	18	101559956668416
16	3	216
17	10	60466176
18	15	470184984576
19	21	21936950640377856
20	26	170581728179578208256
21	3	216

Nim[edit]

Library: bignum
import strformat, strutils
import bignum

var toFind = {0..21}
var results: array[0..21, (int, string)]
var p = newInt(1)
var k = 0
while toFind.card > 0:
  let str = $p
  for n in toFind:
    if str.find($n) >= 0:
      results[n] = (k, str)
      toFind.excl(n)
  p *= 6
  inc k

echo "Smallest values of k such that 6^k contains n:"
for n, (k, s) in results:
  echo &"{n:2}:  6^{k:<2} = {s}"
Output:
Smallest values of k such that 6^k contains n:
 0:  6^9  = 10077696
 1:  6^0  = 1
 2:  6^3  = 216
 3:  6^2  = 36
 4:  6^6  = 46656
 5:  6^6  = 46656
 6:  6^1  = 6
 7:  6^5  = 7776
 8:  6^12 = 2176782336
 9:  6^4  = 1296
10:  6^9  = 10077696
11:  6^16 = 2821109907456
12:  6^4  = 1296
13:  6^13 = 13060694016
14:  6^28 = 6140942214464815497216
15:  6^18 = 101559956668416
16:  6^3  = 216
17:  6^10 = 60466176
18:  6^15 = 470184984576
19:  6^21 = 21936950640377856
20:  6^26 = 170581728179578208256
21:  6^3  = 216

Pascal[edit]

Works with: Free Pascal

Doing long multiplikation like in primorial task.
I used to check every numberstring one after the other on one 6^ n string.Gets really slow on high n
After a closer look into Smallest_power_of_6_whose_decimal_expansion_contains_n#Phix I applied a slghtly modified version of Pete, to get down < 10 secs on my 2200G for DIGITS = 7.TIO.RUN is slower.

program PotOf6;
//First occurence of a numberstring with max DIGTIS digits in 6^n
{$IFDEF FPC}
  {$MODE DELPHI}
  {$Optimization ON,ALL}
{$ELSE}
   {$APPTYPE CONSOLE}
{$ENDIF}

uses
  sysutils;
const
 POT_LIMIT = 70000;
{ DIGITS = 8;
     67584  99999998  46238296
Max power 68479
Found: 100000000 Time used  148.584 secs}
  DIGITS = 7;
type
  tMulElem = Uint32;
  tMul = array of tMulElem;
  tpMul = pUint32;
  tPotArrN = array[0..1] of tMul;

  tFound = record
             foundIndex: Uint32;
             foundStr :Ansistring;
           end;

var
  PotArrN   : tPotArrN;
  Pot_N_str : AnsiString;
  Str_Found : array of tFound;
  FirstMissing :NativeInt;
  T0 : INt64;

procedure Init_Mul(number:NativeInt);
var
  MaxMulIdx : NativeInt;
Begin
  MaxMulIdx := trunc(POT_LIMIT*ln(number)/ln(10)/9+2);
  setlength(PotArrN[0],MaxMulIdx);
  setlength(PotArrN[1],MaxMulIdx);
  PotArrN[0,0] := 1;
end;

function Mul_N(var Mul1,Mul2:tMul;limit,n:Uint32):NativeInt;
//Mul2 = n*Mul1. n must be < LongWordDec !
const
  LongWordDec = 1000*1000*1000;
var
  pM1,pM2 : tpMul;
  carry,prod : Uint64;
begin
  pM1 := @Mul1[0];
  pM2 := @Mul2[0];
  carry := 0;
  result :=0;
  repeat
    prod  := n*pM1[result]+Carry;
    Carry := prod Div LongWordDec;
    pM2[result] := Prod - Carry*LongWordDec;
    inc(result);
  until result > limit;
  IF Carry <> 0 then
    pM2[result] := Carry
  else
    dec(result);
end;

function Commatize(const s: AnsiString):AnsiString;
var
   fromIdx,toIdx :Int32;
Begin
  result := '';
  fromIdx := length(s);
  toIdx := fromIdx-1;
  if toIdx < 3 then
  Begin
    result := s;
    exit;
  end;
  toIdx := 4*(toIdx DIV 3)+toIdx MOD 3;
  inc(toIdx);
  setlength(result,toIdx);
  repeat
    result[toIdx] := s[FromIdx];
    result[toIdx-1] := s[FromIdx-1];
    result[toIdx-2] := s[FromIdx-2];
    result[toIdx-3] := ',';
    dec(toIdx,4);
    dec(FromIdx,3);
  until FromIdx<=3;
  while fromIdx>=1 do
  Begin
    result[toIdx] := s[FromIdx];
    dec(toIdx);
    dec(fromIdx);
  end;
end;

procedure ConvToStr(var s:Ansistring;const Mul:tMul;i:NativeInt);
var
  s9: string[9];
  pS : pChar;
  j,k : NativeInt;
begin
//  i := High(MUL);
  j := (i+1)*9;
  setlength(s,j+1);
  pS := pChar(s);
  // fill complete with '0'
  fillchar(pS[0],j,'0');
  str(Mul[i],S9);
  j := length(s9);
  move(s9[1],pS[0],j);
  k := j;
  dec(i);
  If i >= 0 then
    repeat
      str(Mul[i],S9);// no leading '0'
      j := length(s9);
      inc(k,9);
      //move to the right place, leading '0' is already there
      move(s9[1],pS[k-j],j);
      dec(i);
    until i<0;
  setlength(s,k);
end;

function CheckOneString(const s:Ansistring;pow:NativeInt):NativeInt;
//check every possible number from one to DIGITS digits, 
//if it is still missing in the list
var
  i,k,lmt,num : NativeInt;
  cs : Ansistring;
begin
  result := 0;
  cs := '';
  lmt := length(s);
  For i := 1 to lmt do
  Begin
    k := i;
    num := 0;
    repeat
      num := num*10+ Ord(s[k])-Ord('0');
      IF (num >= FirstMissing) AND (str_Found[num].foundIndex = 0) then
      begin
        str_Found[num].foundIndex:= pow+1;
        // commatize only once. reference counted string
        if cs ='' then
          cs := Commatize(s);
        str_Found[num].foundStr:= cs;
        inc(result);
        if num =irstMissing then
          while str_Found[FirstMissing].foundIndex <> 0 do
            inc(FirstMissing);
      end;
      inc(k)
    until (k>lmt) or (k-i >DIGITS-1);
  end;
end;

var
  i,j,number,toggle,MaxMulIdx,found,decLimit: Int32;
Begin
  T0 := GetTickCount64;
  number := 6;//<1e9 no power of 10 ;-)
  decLimit := 1;
  For i := 1 to digits do
    decLimit *= 10;
  setlength(Str_Found,decLimit);
  Init_Mul(number);

  toggle := 0;
  found := 0;
  FirstMissing := 0;
  MaxMulIdx := 0;
  For j := 0 to POT_LIMIT do
  Begin
    ConvToStr(Pot_N_str,PotArrN[toggle],MaxMulIdx);
    inc(found,CheckOneString(Pot_N_str,j));
    MaxMulIdx := Mul_N(PotArrN[toggle],PotArrN[1-toggle],MaxMulIdx,number);
    toggle := 1-toggle;
    if found>=decLimit then
    Begin
      writeln(#10,'Max power ',j);
      break;
    end;
    if (j and 1023) = 0 then
      write(j:10,found:10,firstMissing:10,#13);
  end;

  writeln(#10,'Found: ',found,' Time used ',(GetTickCount64-T0)/1000:8:3,' secs');
  For i := 0 to 22 do//decLimit-1 do
    with Str_Found[i] do
      if foundIndex >0  then
        writeln(i:10,' ',number,'^',foundIndex-1:5,' ',foundStr);
  readln;
end.
Output:
TIO.RUN output
//Power     found      first missing
         0         1         0
      1024    751817     10020
      2048   2168981    100017
      3072   3733971    100017
      4096   5305316    100672
      5120   6747391    104835
      6144   7922626    575115
      7168   8776137   1000007
      8192   9336696   1000015
      9216   9667898   1000020
     10240   9846933   1000088
     11264   9935108   1000135
     12288   9974783   1000204
     13312   9990953   1000204
     14336   9997035   1000204
     15360   9999102   1000204
     16384   9999744   1029358
     17408   9999934   1029358
     18432   9999978   1029358
     19456   9999997   8091358
     20480   9999999   8091358
     21504   9999999   8091358
Max power 21798

Found: 10000000 Time used   14.882 secs
         0 6^    9 10,077,696
         1 6^    0 1
         2 6^    3 216
         3 6^    2 36
         4 6^    6 46,656
         5 6^    6 46,656
         6 6^    1 6
         7 6^    5 7,776
         8 6^   12 2,176,782,336
         9 6^    4 1,296
        10 6^    9 10,077,696
        11 6^   16 2,821,109,907,456
        12 6^    4 1,296
        13 6^   13 13,060,694,016
        14 6^   28 6,140,942,214,464,815,497,216
        15 6^   18 101,559,956,668,416
        16 6^    3 216
        17 6^   10 60,466,176
        18 6^   15 470,184,984,576
        19 6^   21 21,936,950,640,377,856
        20 6^   26 170,581,728,179,578,208,256
        21 6^    3 216
        22 6^   22 131,621,703,842,267,136

Real time: 15.373 s
User time: 14.953 s
Sys. time: 0.254 s
CPU share: 98.92 %

Perl[edit]

use strict;
use warnings;
use List::Util 'first';
use Math::AnyNum ':overload';

sub comma { reverse ((reverse shift) =~ s/(.{3})/$1,/gr) =~ s/^,//r }

for my $n (0..21, 314159) {
    my $e = first { 6**$_ =~ /$n/ } 0..1000;
    printf "%7d:  6^%-3s  %s\n", $n, $e, comma 6**$e;
}
Output:
      0:  6^9    10,077,696
      1:  6^0    1
      2:  6^3    216
      3:  6^2    36
      4:  6^6    46,656
      5:  6^6    46,656
      6:  6^1    6
      7:  6^5    7,776
      8:  6^12   2,176,782,336
      9:  6^4    1,296
     10:  6^9    10,077,696
     11:  6^16   2,821,109,907,456
     12:  6^4    1,296
     13:  6^13   13,060,694,016
     14:  6^28   6,140,942,214,464,815,497,216
     15:  6^18   101,559,956,668,416
     16:  6^3    216
     17:  6^10   60,466,176
     18:  6^15   470,184,984,576
     19:  6^21   21,936,950,640,377,856
     20:  6^26   170,581,728,179,578,208,256
     21:  6^3    216
 314159:  6^494  2,551,042,473,957,557,281,758,472,595,966,885,638,262,058,644,568,332,160,010,313,393,465,384,231,415,969,801,503,269,402,221,368,959,426,761,447,049,526,922,498,341,120,174,041,236,629,812,681,424,262,988,020,546,286,492,213,224,906,594,147,652,459,693,833,191,626,748,973,370,777,591,205,509,673,825,541,899,874,436,305,798,094,943,728,762,682,333,192,202,041,960,669,401,031,964,634,164,426,985,990,195,192,836,400,994,016,666,910,919,499,884,972,133,471,176,804,190,463,444,807,178,864,658,551,422,631,018,496

Phix[edit]

Another good opportunity to do some string math, this time with embedded commas. Scales effortlessly.
(Related recent task: Show_the_(decimal)_value_of_a_number_of_1s_appended_with_a_3,_then_squared#Phix)

constant lim = 22           -- (tested to 10,000,000)
atom t0 = time(), t1 = t0+1
sequence res = repeat(0,lim),
         pwr = repeat(0,lim)
string p6 = "1"
res[2] = p6
integer found = 1, p = 0
while found<lim do
    integer carry = 0
    for i=length(p6) to 1 by -1 do
        if p6[i]!=',' then
            integer digit = (p6[i]-'0')*6+carry
            p6[i] = remainder(digit,10)+'0'
            carry = floor(digit/10)
        end if
    end for
    if carry then
        if remainder(length(p6)+1,4)=0 then
            p6 = "," & p6
        end if
        p6 = carry+'0' & p6
    end if
    p += 1
    for i=1 to length(p6) do
        if p6[i]!=',' then
            integer digit = 0, j = i
            while j<=length(p6) and digit<=lim do
                j += p6[j]=','
                digit = digit*10+p6[j]-'0'
                if digit<lim and res[digit+1]=0 then
                    res[digit+1] = p6
                    pwr[digit+1] = p
                    found += 1
                end if
                j += 1
            end while
        end if
    end for
    if time()>t1 then
        progress("found %,d/%,d, at 6^%,d which has %,d digits (%s)",
                 {found,lim,p,length(p6)*3/4,elapsed(time()-t0)})
        t1 = time()+1
    end if
end while
papply(true,printf,{1,{"%2d  %29s = 6^%d\n"},shorten(columnize({tagset(lim-1,0),res,pwr}),"",10)})
Output:
 0                     10,077,696 = 6^9
 1                              1 = 6^0
 2                            216 = 6^3
 3                             36 = 6^2
 4                         46,656 = 6^6
 5                         46,656 = 6^6
 6                              6 = 6^1
 7                          7,776 = 6^5
 8                  2,176,782,336 = 6^12
 9                          1,296 = 6^4
10                     10,077,696 = 6^9
11              2,821,109,907,456 = 6^16
12                          1,296 = 6^4
13                 13,060,694,016 = 6^13
14  6,140,942,214,464,815,497,216 = 6^28
15            101,559,956,668,416 = 6^18
16                            216 = 6^3
17                     60,466,176 = 6^10
18                470,184,984,576 = 6^15
19         21,936,950,640,377,856 = 6^21
20    170,581,728,179,578,208,256 = 6^26
21                            216 = 6^3

A limit of 10,000,000 takes 1 min 41s, reaches 6^21,798 which has 16,963 digits (not including commas) and is the first to contain 8091358, at offset 13,569.

Python[edit]

def smallest_six(n):
    p = 1
    while str(n) not in str(p): p *= 6
    return p
    
for n in range(22):
    print("{:2}: {}".format(n, smallest_six(n)))
Output:
 0: 10077696
 1: 1
 2: 216
 3: 36
 4: 46656
 5: 46656
 6: 6
 7: 7776
 8: 2176782336
 9: 1296
10: 10077696
11: 2821109907456
12: 1296
13: 13060694016
14: 6140942214464815497216
15: 101559956668416
16: 216
17: 60466176
18: 470184984576
19: 21936950640377856
20: 170581728179578208256
21: 216

Raku[edit]

use Lingua::EN::Numbers;

my @po6 = ^Inf .map: *.exp: 6;

put join "\n", (flat ^22, 120).map: -> $n {
    sprintf "%3d: 6%-4s %s", $n, .&super, comma @po6[$_]
    given @po6.first: *.contains($n), :k
};
Output:
  0: 6⁹    10,077,696
  1: 6⁰    1
  2: 6³    216
  3: 6²    36
  4: 6⁶    46,656
  5: 6⁶    46,656
  6: 6¹    6
  7: 6⁵    7,776
  8: 6¹²   2,176,782,336
  9: 6⁴    1,296
 10: 6⁹    10,077,696
 11: 6¹⁶   2,821,109,907,456
 12: 6⁴    1,296
 13: 6¹³   13,060,694,016
 14: 6²⁸   6,140,942,214,464,815,497,216
 15: 6¹⁸   101,559,956,668,416
 16: 6³    216
 17: 6¹⁰   60,466,176
 18: 6¹⁵   470,184,984,576
 19: 6²¹   21,936,950,640,377,856
 20: 6²⁶   170,581,728,179,578,208,256
 21: 6³    216
120: 6¹⁴⁷  2,444,746,349,972,956,194,083,608,044,935,243,159,422,957,210,683,702,349,648,543,934,214,737,968,217,920,868,940,091,707,112,078,529,114,392,164,827,136

REXX[edit]

/*REXX pgm finds the smallest (decimal) power of  6  which contains  N,  where  N < 22. */
numeric digits 100                               /*ensure enough decimal digs for  6**N */
parse arg hi .                                   /*obtain optional argument from the CL.*/
if hi=='' | hi==","  then hi= 22                 /*Not specified?  Then use the default.*/
w= 50                                            /*width of a number in any column.     */
               @smp6= ' smallest power of  six  (expressed in decimal)  which contains  N'
say '  N  │ power │'center(@smp6, 20 + w     )   /*display the   title   of the output. */
say '─────┼───────┼'center(""   , 20 + w, '─')   /*   "     "  separator  "  "     "    */

      do j=0  for hi                             /*look for a power of 6 that contains N*/
                     do p=0;   x= 6**p           /*compute a power of six (in decimal). */
                     if pos(j, x)>0  then leave  /*does the power contain an   N ?      */
                     end   /*p*/
      c= commas(x)                               /*maybe add commas to the powe of six. */
      z= right(c, max(w, length(c) ) )           /*show a power of six, allow biger #s. */
      say center(j, 5)'│'center(p, 7)"│"   z     /*display what we have so far  (cols). */
      end   /*j*/

say '─────┴───────┴'center(""   , 20 + w, '─')   /*   "     "  separator  "  "     "    */
exit 0                                           /*stick a fork in it,  we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
commas: parse arg ?;  do jc=length(?)-3  to 1  by -3; ?=insert(',', ?, jc); end;  return ?
output   when using the default input:
  N  │ power │   smallest power of  six  (expressed in decimal)  which contains  N
─────┼───────┼──────────────────────────────────────────────────────────────────────
  0  │   9   │                                         10,077,696
  1  │   0   │                                                  1
  2  │   3   │                                                216
  3  │   2   │                                                 36
  4  │   6   │                                             46,656
  5  │   6   │                                             46,656
  6  │   1   │                                                  6
  7  │   5   │                                              7,776
  8  │  12   │                                      2,176,782,336
  9  │   4   │                                              1,296
 10  │   9   │                                         10,077,696
 11  │  16   │                                  2,821,109,907,456
 12  │   4   │                                              1,296
 13  │  13   │                                     13,060,694,016
 14  │  28   │                      6,140,942,214,464,815,497,216
 15  │  18   │                                101,559,956,668,416
 16  │   3   │                                                216
 17  │  10   │                                         60,466,176
 18  │  15   │                                    470,184,984,576
 19  │  21   │                             21,936,950,640,377,856
 20  │  26   │                        170,581,728,179,578,208,256
 21  │   3   │                                                216
─────┴───────┴──────────────────────────────────────────────────────────────────────

Ring[edit]

load "stdlib.ring"

decimals(0)
see "working..." + nl
see "Smallest power of 6 whose decimal expansion contains n:" + nl

num = 0
limit = 200

for n = 1 to 21
    strn = string(n)
    for m = 0 to limit
        strpow = string(pow(6,m))
        ind = substr(strpow,strn)
        if ind > 0
           see "" + n + ". " + "6^" + m + " = " + strpow + nl
           exit
        ok
    next
next

see "done..." + nl
Output:
working...
Smallest power of 6 whose decimal expansion contains n:
1. 6^0 = 1
2. 6^3 = 216
3. 6^2 = 36
4. 6^6 = 46656
5. 6^6 = 46656
6. 6^1 = 6
7. 6^5 = 7776
8. 6^12 = 2176782336
9. 6^4 = 1296
10. 6^9 = 10077696
11. 6^16 = 2821109907456
12. 6^4 = 1296
13. 6^13 = 13060694016
14. 6^28 = 6140942214464815497216
15. 6^18 = 101559956668416
16. 6^3 = 216
17. 6^10 = 60466176
18. 6^15 = 470184984576
19. 6^21 = 21936950640377856
20. 6^26 = 170581728179578208256
21. 6^3 = 216
done...

Wren[edit]

Library: Wren-big
Library: Wren-fmt
import "/big" for BigInt
import "/fmt" for Fmt

System.print(" n  smallest power of 6 which contains n")
var six = BigInt.new(6)
for (n in 0..21) {
    var i = 0
    while (true) {
        var pow6 = six.pow(i).toString
        if (pow6.contains(n.toString)) {
            Fmt.print("$2d  6^$-2d = $,s", n, i, pow6)
            break
        }
        i = i + 1
    }
}
Output:
 n  smallest power of 6 which contains n
 0  6^9  = 10,077,696
 1  6^0  = 1
 2  6^3  = 216
 3  6^2  = 36
 4  6^6  = 46,656
 5  6^6  = 46,656
 6  6^1  = 6
 7  6^5  = 7,776
 8  6^12 = 2,176,782,336
 9  6^4  = 1,296
10  6^9  = 10,077,696
11  6^16 = 2,821,109,907,456
12  6^4  = 1,296
13  6^13 = 13,060,694,016
14  6^28 = 6,140,942,214,464,815,497,216
15  6^18 = 101,559,956,668,416
16  6^3  = 216
17  6^10 = 60,466,176
18  6^15 = 470,184,984,576
19  6^21 = 21,936,950,640,377,856
20  6^26 = 170,581,728,179,578,208,256
21  6^3  = 216

Yabasic[edit]

Translation of: Python
// Rosetta Code problem: http://rosettacode.org/wiki/Smallest_power_of_6_whose_decimal_expansion_contains_n
// by Galileo, 05/2022

sub smallest_six(n)
    local p, n$
    
    n$ = str$(n)
    p = 1
    while not instr(str$(p, "%1.f"), n$) p = p * 6 : wend
    return p
end sub
 
for n = 0 to 21 : print n, ": ", str$(smallest_six(n), "%1.f") : next
Output:
0: 10077696
1: 1
2: 216
3: 36
4: 46656
5: 46656
6: 6
7: 7776
8: 2176782336
9: 1296
10: 10077696
11: 2821109907456
12: 1296
13: 13060694016
14: 6140942214464815497216
15: 101559956668416
16: 216
17: 60466176
18: 470184984576
19: 21936950640377856
20: 170581728179578208256
21: 216
---Program done, press RETURN---