Super-d numbers

From Rosetta Code
Task
Super-d numbers
You are encouraged to solve this task according to the task description, using any language you may know.

A super-d number is a positive, decimal (base ten) integer n such that d × n^d has at least d consecutive digits d where

   2 ≤ d ≤ 9

For instance, 753 is a super-3 number because 3 × 753^3 = 1280873331.


Super-d   numbers are also shown on   MathWorld™   as   super-d   or   super-d.


Task
  • Write a function/procedure/routine to find super-d numbers.
  • For   d=2   through   d=6,   use the routine to show the first   10   super-d numbers.


Extra credit
  • Show the first   10   super-7, super-8, and/or super-9 numbers.   (Optional)


See also


F#[edit]

The Function
 
// Generate Super-N numbers. Nigel Galloway: October 12th., 2019
let superD N=
let I=bigint(pown 10 N)
let G=bigint N
let E=G*(111111111I%I)
let rec fL n=match (E-n%I).IsZero with true->true |_->if (E*10I)<n then false else fL (n/10I)
seq{1I..999999999999999999I}|>Seq.choose(fun n->if fL (G*n**N) then Some n else None)
 
The Task
 
superD 2 |> Seq.take 10 |> Seq.iter(printf "%A "); printfn ""
superD 3 |> Seq.take 10 |> Seq.iter(printf "%A "); printfn ""
superD 4 |> Seq.take 10 |> Seq.iter(printf "%A "); printfn ""
superD 5 |> Seq.take 10 |> Seq.iter(printf "%A "); printfn ""
superD 6 |> Seq.take 10 |> Seq.iter(printf "%A "); printfn ""
superD 7 |> Seq.take 10 |> Seq.iter(printf "%A "); printfn ""
superD 8 |> Seq.take 10 |> Seq.iter(printf "%A "); printfn ""
superD 9 |> Seq.take 10 |> Seq.iter(printf "%A "); printfn ""
 
Output:
19 31 69 81 105 106 107 119 127 131
261 462 471 481 558 753 1036 1046 1471 1645
1168 4972 7423 7752 8431 10267 11317 11487 11549 11680
4602 5517 7539 12955 14555 20137 20379 26629 32767 35689
27257 272570 302693 323576 364509 502785 513675 537771 676657 678146
140997 490996 1184321 1259609 1409970 1783166 1886654 1977538 2457756 2714763
185423 641519 1551728 1854230 6415190 12043464 12147605 15517280 16561735 18542300
17546133 32613656 93568867 107225764 109255734 113315082 121251742 175461330 180917907 182557181

Factor[edit]

USING: arrays formatting io kernel lists lists.lazy math
math.functions math.ranges math.text.utils prettyprint sequences
;
IN: rosetta-code.super-d
 
: super-d? ( seq n d -- ? ) tuck ^ * 1 digit-groups subseq? ;
 
: super-d ( d -- list )
[ dup <array> ] [ drop 1 lfrom ] [ ] tri [ super-d? ] curry
with lfilter ;
 
: super-d-demo ( -- )
10 2 6 [a,b] [
dup "First 10 super-%d numbers:\n" printf
super-d ltake list>array [ pprint bl ] each nl nl
] with each ;
 
MAIN: super-d-demo
Output:
First 10 super-2 numbers:
19 31 69 81 105 106 107 119 127 131 

First 10 super-3 numbers:
261 462 471 481 558 753 1036 1046 1471 1645 

First 10 super-4 numbers:
1168 4972 7423 7752 8431 10267 11317 11487 11549 11680 

First 10 super-5 numbers:
4602 5517 7539 12955 14555 20137 20379 26629 32767 35689 

First 10 super-6 numbers:
27257 272570 302693 323576 364509 502785 513675 537771 676657 678146 

Fōrmulæ[edit]

In this page you can see the solution of this task.

Fōrmulæ programs are not textual, visualization/edition of programs is done showing/manipulating structures but not text (more info). Moreover, there can be multiple visual representations of the same program. Even though it is possible to have textual representation —i.e. XML, JSON— they are intended for transportation effects more than visualization and edition.

The option to show Fōrmulæ programs and their results is showing images. Unfortunately images cannot be uploaded in Rosetta Code.

Go[edit]

Simple brute force approach and so not particularly quick - about 2.25 minutes on a Core i7.

package main
 
import (
"fmt"
"math/big"
"strings"
"time"
)
 
func main() {
start := time.Now()
rd := []string{"22", "333", "4444", "55555", "666666", "7777777", "88888888", "999999999"}
one := big.NewInt(1)
nine := big.NewInt(9)
for i := big.NewInt(2); i.Cmp(nine) <= 0; i.Add(i, one) {
fmt.Printf("First 10 super-%d numbers:\n", i)
ii := i.Uint64()
k := new(big.Int)
count := 0
inner:
for j := big.NewInt(3); ; j.Add(j, one) {
k.Exp(j, i, nil)
k.Mul(i, k)
ix := strings.Index(k.String(), rd[ii-2])
if ix >= 0 {
count++
fmt.Printf("%d ", j)
if count == 10 {
fmt.Printf("\nfound in %d ms\n\n", time.Since(start).Milliseconds())
break inner
}
}
}
}
}
Output:
First 10 super-2 numbers:
19 31 69 81 105 106 107 119 127 131 
found in 0 ms

First 10 super-3 numbers:
261 462 471 481 558 753 1036 1046 1471 1645 
found in 1 ms

First 10 super-4 numbers:
1168 4972 7423 7752 8431 10267 11317 11487 11549 11680 
found in 7 ms

First 10 super-5 numbers:
4602 5517 7539 12955 14555 20137 20379 26629 32767 35689 
found in 28 ms

First 10 super-6 numbers:
27257 272570 302693 323576 364509 502785 513675 537771 676657 678146 
found in 285 ms

First 10 super-7 numbers:
140997 490996 1184321 1259609 1409970 1783166 1886654 1977538 2457756 2714763 
found in 1517 ms

First 10 super-8 numbers:
185423 641519 1551728 1854230 6415190 12043464 12147605 15517280 16561735 18542300 
found in 11117 ms

First 10 super-9 numbers:
17546133 32613656 93568867 107225764 109255734 113315082 121251742 175461330 180917907 182557181 
found in 135616 ms

Julia[edit]

Translation of: Phix
function superd(N)
println("First 10 super-$N numbers:")
count, j = 0, BigInt(3)
target = Char('0' + N)^N
while count < 10
if occursin(target, string(j^N * N))
count += 1
print("$j ")
end
j += 1
end
println()
end
 
for n in 2:9
@time superd(n)
end
 
Output:
First 10 super-2 numbers:
19 31 69 81 105 106 107 119 127 131
  0.017720 seconds (32.80 k allocations: 1.538 MiB)
First 10 super-3 numbers:
261 462 471 481 558 753 1036 1046 1471 1645
  0.003976 seconds (47.33 k allocations: 985.352 KiB)
First 10 super-4 numbers:
1168 4972 7423 7752 8431 10267 11317 11487 11549 11680
  0.018958 seconds (327.37 k allocations: 6.808 MiB)
First 10 super-5 numbers:
4602 5517 7539 12955 14555 20137 20379 26629 32767 35689
  0.060683 seconds (1.02 M allocations: 21.561 MiB)
First 10 super-6 numbers:
27257 272570 302693 323576 364509 502785 513675 537771 676657 678146
  1.395905 seconds (19.14 M allocations: 419.551 MiB, 19.77% gc time)
First 10 super-7 numbers:
140997 490996 1184321 1259609 1409970 1783166 1886654 1977538 2457756 2714763
  5.604611 seconds (77.81 M allocations: 1.687 GiB, 18.66% gc time)
First 10 super-8 numbers:
185423 641519 1551728 1854230 6415190 12043464 12147605 15517280 16561735 18542300
 38.827266 seconds (539.13 M allocations: 12.106 GiB, 19.11% gc time)
First 10 super-9 numbers:
17546133 32613656 93568867 107225764 109255734 113315082 121251742 175461330 180917907 182557181
380.576442 seconds (5.26 G allocations: 124.027 GiB, 18.02% gc time)

Pascal[edit]

Works with: Free Pascal

gmp is fast.Same brute force as Go

program Super_D;
uses
sysutils,gmp;
 
var
s :ansistring;
s_comp : ansistring;
test : mpz_t;
i,j,dgt,cnt : NativeUint;
Begin
mpz_init(test);
 
for dgt := 2 to 9 do
Begin
//create '22' to '999999999'
i := dgt;
For j := 2 to dgt do
i := i*10+dgt;
s_comp := IntToStr(i);
writeln('Finding ',s_comp,' in ',dgt,'*i**',dgt);
 
i := dgt;
cnt := 0;
repeat
mpz_ui_pow_ui(test,i,dgt);
mpz_mul_ui(test,test,dgt);
setlength(s,mpz_sizeinbase(test,10));
mpz_get_str(pChar(s),10,test);
IF Pos(s_comp,s) <> 0 then
Begin
write(i,' ');
inc(cnt);
end;
inc(i);
until cnt = 10;
writeln;
end;
mpz_clear(test);
End.
Output:
Finding 22 in 2*i**2
19 31 69 81 105 106 107 119 127 131 
Finding 333 in 3*i**3
261 462 471 481 558 753 1036 1046 1471 1645 
Finding 4444 in 4*i**4
1168 4972 7423 7752 8431 10267 11317 11487 11549 11680 
Finding 55555 in 5*i**5
4602 5517 7539 12955 14555 20137 20379 26629 32767 35689 
Finding 666666 in 6*i**6
27257 272570 302693 323576 364509 502785 513675 537771 676657 678146 
Finding 7777777 in 7*i**7
140997 490996 1184321 1259609 1409970 1783166 1886654 1977538 2457756 2714763 
Finding 88888888 in 8*i**8
185423 641519 1551728 1854230 6415190 12043464 12147605 15517280 16561735 18542300 
Finding 999999999 in 9*i**9
17546133 32613656 93568867 107225764 109255734 113315082 121251742 175461330 180917907 182557181 
real	1m11,239s

//only calc 9*i**9 [2..182557181]
Finding 999999999 in 9*i**9
real    0m7,577s
//calc 9*i**9 [2..182557181] and convert to String with preset length 100 -> takes longer to find '999999999'
real    0m40,094s
//calc 9*i**9 [2..182557181] and convert to String and setlength
real    0m41,358s
//complete task with finding
Finding 999999999 in 9*i**9
17546133 32613656 93568867 107225764 109255734 113315082 121251742 175461330 180917907 182557181 
real	1m5,134s

Perl[edit]

use strict;
use warnings;
use bigint;
use feature 'say';
 
sub super {
my $d = shift;
my $run = $d x $d;
my @super;
my $i = 0;
my $n = 0;
while ( $i < 10 ) {
if (index($n ** $d * $d, $run) > -1) {
push @super, $n;
++$i;
}
++$n;
}
@super;
}
 
say "\nFirst 10 super-$_ numbers:\n", join ' ', super($_) for 2..6;
First 10 super-2 numbers:
19 31 69 81 105 106 107 119 127 131

First 10 super-3 numbers:
261 462 471 481 558 753 1036 1046 1471 1645

First 10 super-4 numbers:
1168 4972 7423 7752 8431 10267 11317 11487 11549 11680

First 10 super-5 numbers:
4602 5517 7539 12955 14555 20137 20379 26629 32767 35689

First 10 super-6 numbers:
27257 272570 302693 323576 364509 502785 513675 537771 676657 678146

Perl 6[edit]

Works with: Rakudo version 2019.07.1

2 - 6 take a few seconds, 7 about 17 seconds, 8 about 90... 9, bleh... around 700 seconds.

sub super (\d) {
my \run = d x d;
^.hyper.grep: -> \n { (d * n ** d).Str.contains: run }
}
 
(2..9).race(:1batch).map: {
my $now = now;
put "\nFirst 10 super-$_ numbers:\n{.&super[^10]}\n{(now - $now).round(.1)} sec."
}
First 10 super-2 numbers:
19 31 69 81 105 106 107 119 127 131
0.1 sec.

First 10 super-3 numbers:
261 462 471 481 558 753 1036 1046 1471 1645
0.1 sec.

First 10 super-4 numbers:
1168 4972 7423 7752 8431 10267 11317 11487 11549 11680
0.3 sec.

First 10 super-5 numbers:
4602 5517 7539 12955 14555 20137 20379 26629 32767 35689
0.6 sec.

First 10 super-6 numbers:
27257 272570 302693 323576 364509 502785 513675 537771 676657 678146
5.2 sec.

First 10 super-7 numbers:
140997 490996 1184321 1259609 1409970 1783166 1886654 1977538 2457756 2714763
17.1 sec.

First 10 super-8 numbers:
185423 641519 1551728 1854230 6415190 12043464 12147605 15517280 16561735 18542300
92.1 sec.

First 10 super-9 numbers:
17546133 32613656 93568867 107225764 109255734 113315082 121251742 175461330 180917907 182557181
704.7 sec.

Phix[edit]

Translation of: Go
include mpfr.e
 
procedure main()
atom t0 = time()
mpz k = mpz_init()
for i=2 to 9 do
printf(1,"First 10 super-%d numbers:\n", i)
integer count := 0, j = 3
string tgt = repeat('0'+i,i)
while count<10 do
mpz_ui_pow_ui(k,j,i)
mpz_mul_si(k,k,i)
string s = mpz_get_str(k)
integer ix = match(tgt,s)
if ix then
count += 1
printf(1,"%d ", j)
end if
j += 1
end while
printf(1,"\nfound in %s\n\n", {elapsed(time()-t0)})
end for
end procedure
main()
Output:
First 10 super-2 numbers:
19 31 69 81 105 106 107 119 127 131
found in 0.0s

First 10 super-3 numbers:
261 462 471 481 558 753 1036 1046 1471 1645
found in 0.0s

First 10 super-4 numbers:
1168 4972 7423 7752 8431 10267 11317 11487 11549 11680
found in 0.2s

First 10 super-5 numbers:
4602 5517 7539 12955 14555 20137 20379 26629 32767 35689
found in 0.5s

First 10 super-6 numbers:
27257 272570 302693 323576 364509 502785 513675 537771 676657 678146
found in 6.8s

First 10 super-7 numbers:
140997 490996 1184321 1259609 1409970 1783166 1886654 1977538 2457756 2714763
found in 33.2s

First 10 super-8 numbers:
185423 641519 1551728 1854230 6415190 12043464 12147605 15517280 16561735 18542300
found in 3 minutes and 47s

First 10 super-9 numbers:
17546133 <killed>

Python[edit]

from itertools import islice, count
 
def superd(d):
if d != int(d) or not 2 <= d <= 9:
raise ValueError("argument must be integer from 2 to 9 inclusive")
tofind = str(d) * d
for n in count(2):
if tofind in str(d * n ** d):
yield n
 
if __name__ == '__main__':
for d in range(2, 8):
print(f"{d}:", ', '.join(str(n) for n in islice(superd(d), 10)))
Output:
2: 19, 31, 69, 81, 105, 106, 107, 119, 127, 131
3: 261, 462, 471, 481, 558, 753, 1036, 1046, 1471, 1645
4: 1168, 4972, 7423, 7752, 8431, 10267, 11317, 11487, 11549, 11680
5: 4602, 5517, 7539, 12955, 14555, 20137, 20379, 26629, 32767, 35689
6: 27257, 272570, 302693, 323576, 364509, 502785, 513675, 537771, 676657, 678146
7: 140997, 490996, 1184321, 1259609, 1409970, 1783166, 1886654, 1977538, 2457756, 2714763
8: 185423, 641519, 1551728, 1854230, 6415190, 12043464, 12147605, 15517280, 16561735, 18542300

REXX[edit]

/*REXX program computes and displays the first  N  super─d  numbers for D from LO to HI.*/
numeric digits 100 /*ensure enough decimal digs for calc. */
parse arg n LO HI . /*obtain optional arguments from the CL*/
if n=='' | n=="," then n= 10 /*the number of super─d numbers to calc*/
if LO=='' | LO=="," then LO= 2 /*low end of D for the super─d nums.*/
if HI=='' | HI=="," then HI= 9 /*high " " " " " " " */
/* [↓] process D from LO ──► HI. */
do d=LO to HI; #= 0; $= /*count & list of super─d nums (so far)*/
z= copies(d, d) /*the string that is being searched for*/
do j=2 until #==n /*search for super─d numbers 'til found*/
if pos(z, d * j**d)==0 then iterate /*does product have the required reps? */
#= # + 1; $= $ j /*bump counter; add the number to list*/
end /*j*/
say
say center(' the first ' n " super-"d 'numbers ', digits(), "═")
say $
end /*d*/ /*stick a fork in it, we're all done. */
output   when using the default inputs:
══════════════════════════════════ the first  10  super-2 numbers ══════════════════════════════════
 19 31 69 81 105 106 107 119 127 131

══════════════════════════════════ the first  10  super-3 numbers ══════════════════════════════════
 261 462 471 481 558 753 1036 1046 1471 1645

══════════════════════════════════ the first  10  super-4 numbers ══════════════════════════════════
 1168 4972 7423 7752 8431 10267 11317 11487 11549 11680

══════════════════════════════════ the first  10  super-5 numbers ══════════════════════════════════
 4602 5517 7539 12955 14555 20137 20379 26629 32767 35689

══════════════════════════════════ the first  10  super-6 numbers ══════════════════════════════════
 27257 272570 302693 323576 364509 502785 513675 537771 676657 678146

══════════════════════════════════ the first  10  super-7 numbers ══════════════════════════════════
 140997 490996 1184321 1259609 1409970 1783166 1886654 1977538 2457756 2714763

══════════════════════════════════ the first  10  super-8 numbers ══════════════════════════════════
 185423 641519 1551728 1854230 6415190 12043464 12147605 15517280 16561735 18542300

══════════════════════════════════ the first  10  super-9 numbers ══════════════════════════════════
 17546133 32613656 93568867 107225764 109255734 113315082 121251742 175461330 180917907 182557181

zkl[edit]

Library: GMP
GNU Multiple Precision Arithmetic Library
var [const] BI=Import("zklBigNum");  // libGMP
 
fcn superDW(d){
digits:=d.toString()*d;
[2..].tweak('wrap(n)
{ BI(n).pow(d).mul(d).toString().holds(digits) and n or Void.Skip });
}
foreach d in ([2..8]){ println(d," : ",superDW(d).walk(10).concat(" ")) }
Output:
2 : 19 31 69 81 105 106 107 119 127 131
3 : 261 462 471 481 558 753 1036 1046 1471 1645
4 : 1168 4972 7423 7752 8431 10267 11317 11487 11549 11680
5 : 4602 5517 7539 12955 14555 20137 20379 26629 32767 35689
6 : 27257 272570 302693 323576 364509 502785 513675 537771 676657 678146
7 : 140997 490996 1184321 1259609 1409970 1783166 1886654 1977538 2457756 2714763
8 : 185423 641519 1551728 1854230 6415190 12043464 12147605 15517280 16561735 18542300