Super-d numbers

Super-d numbers 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.

A super-d number is a positive 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.

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

• Wolfram MathWorld - Super-d Number
• OEIS: A014569 - Super-3 Numbers

Factor

<lang factor>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</lang>

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

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

Simple brute force approach and so not particularly quick - about 2.25 minutes on a Core i7. <lang go>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
               }
           }
       }
   }

}</lang>

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

Perl

<lang perl>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;</lang>

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

Works with: Rakudo version 2019.07.1

Single-threaded

2 - 6 takes a few seconds, 7 & 8 take a few minutes; I got tired of waiting for 9.

<lang perl6>sub super ($d) {

   my $run = $d x $d;
   "First 10 super-{$d} numbers:\n{ (^∞ .grep: ($d * * ** $d).Str.contains($run) )[^10]}\n"

}

put .&super for 2 .. 8</lang>

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

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

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

Concurrent

Since waiting can be tiresome, make things faster with the race concurrency function. The only problem is that we're not sure what the stopping point is ahead of time, so when it seems safe to quit (calculate a few more result than strictly needed, since the parallel workers don't coordinate), break out of the concurrent block by raising an exception. The usual concurrency caveats apply: @super must be sized ahead of time, and only modified with an atomic operation (⚛++).

Concurrency makes little difference for d < 6, but the benefits accrue rapidly after that (greater than 10-fold speed-up for d = 8, with an 8-core CPU). However, in the end, you'll still have to wait a quite bit or the super-9 values. <lang perl6>sub super-d ($d,$max) {

   my $max-plus = $max + floor 2*$max/$d;
   my @super[2*$max-plus];
   {
       my $digits = $d x $d;
       my $chunk  = 250 * $d;
       my atomicint $found = 0;
       (0, 1*$chunk, 2*$chunk ... *).race(:1batch).map: -> $i {
            @super[$found⚛++] = $_ if ($_ ** $d  *  $d).contains($digits) for 1+$i .. 1+$i+$chunk;
            X::AdHoc.new.throw if $found ≥ $max-plus;
       }
       CATCH { when X::AdHoc { return @super.grep(so *).sort.head($max) } }
   }

}

say "$_: " ~ join ' ', super-d($_,10) for 2..9;</lang>

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
9: 17546133 32613656 93568867 107225764 109255734 113315082 121251742 175461330 180917907 182557181

REXX

<lang rexx>/*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 " " " " " " " */

    do d=LO  for HI-1                           /*process the  D  from  LO  through  HI*/
    #= 0                                        /*count of the super-d numbers (so far)*/
    $=                                          /* list  "  "     "       "      "  "  */
    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. */</lang>

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

Library: GMP

GNU Multiple Precision Arithmetic Library

<lang zkl>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(" ")) }</lang>

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