Rosetta Code

Super-d numbers

Revision as of 17:57, 28 August 2022 by Thundergnat (talk | contribs) (syntax highlighting fixup automation)

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

Task
Super-d numbers
You are encouraged to solve this task according to the task description, using any language you may know.
   2 ≤ d ≤ 9

For instance, 753 is a super-3 number because 3 × 7533 = 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



11l

Translation of: D
V rd = [‘22’, ‘333’, ‘4444’, ‘55555’, ‘666666’, ‘7777777’, ‘88888888’, ‘999999999’]

L(ii) 2..7
   print(‘First 10 super-#. numbers:’.format(ii))
   V count = 0

   BigInt j = 3
   L
      V k = ii * j^ii
      I rd[ii - 2] C String(k)
         count++
         print(j, end' ‘ ’)
         I count == 10
            print("\n")
            L.break
      j++
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

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

C#

With / Without BigInteger

Library: System.Numerics

Done three ways, two with BigIntegers, and one with a UIint64 structure. More details on the "Mostly addition" method on the discussion page. 

using System;
using System.Collections.Generic;
using BI = System.Numerics.BigInteger;
using lbi = System.Collections.Generic.List<System.Numerics.BigInteger[]>;
using static System.Console;

class Program {

    // provides 320 bits (90 decimal digits)
    struct LI { public UInt64 lo, ml, mh, hi, tp; }

    const UInt64 Lm = 1_000_000_000_000_000_000UL;
    const string Fm = "D18";

    static void inc(ref LI d, LI s) { // d += s
        d.lo += s.lo; while (d.lo >= Lm) { d.ml++; d.lo -= Lm; }
        d.ml += s.ml; while (d.ml >= Lm) { d.mh++; d.ml -= Lm; }
        d.mh += s.mh; while (d.mh >= Lm) { d.hi++; d.mh -= Lm; }
        d.hi += s.hi; while (d.hi >= Lm) { d.tp++; d.hi -= Lm; }
        d.tp += s.tp;
    }

    static void set(ref LI d, UInt64 s) { // d = s
        d.lo = s; d.ml = d.mh = d.hi = d.tp = 0;
    }

    const int ls = 10;

    static lbi co = new lbi { new BI[] { 0 } }; // for BigInteger addition
    static List<LI[]> Co = new List<LI[]> { new LI[1] }; // for UInt64 addition

    static Int64 ipow(Int64 bas, Int64 exp) { // Math.Pow()
        Int64 res = 1; while (exp != 0) {
            if ((exp & 1) != 0) res *= bas; exp >>= 1; bas *= bas;
        }
        return res;
    }

    // finishes up, shows performance value
    static void fin() { WriteLine("{0}s", (DateTime.Now - st).TotalSeconds.ToString().Substring(0, 5)); }

    static void funM(int d) { // straightforward BigInteger method, medium performance
        string s = new string(d.ToString()[0], d); Write("{0}: ", d);
        for (int i = 0, c = 0; c < ls; i++)
            if ((BI.Pow((BI)i, d) * d).ToString().Contains(s))
                Write("{0} ", i, ++c);
        fin();
    }

    static void funS(int d) { // BigInteger "mostly adding" method, low performance
        BI[] m = co[d];
        string s = new string(d.ToString()[0], d); Write("{0}: ", d);
        for (int i = 0, c = 0; c < ls; i++) {
            if ((d * m[0]).ToString().Contains(s))
                Write("{0} ", i, ++c);
            for (int j = d, k = d - 1; j > 0; j = k--) m[k] += m[j];
        }
        fin();
    }

    static string scale(uint s, ref LI x) { // performs a small multiply and returns a string value
        ulong Lo = x.lo * s, Ml = x.ml * s, Mh = x.mh * s, Hi = x.hi * s, Tp = x.tp * s;
        while (Lo >= Lm) { Lo -= Lm; Ml++; }
        while (Ml >= Lm) { Ml -= Lm; Mh++; }
        while (Mh >= Lm) { Mh -= Lm; Hi++; }
        while (Hi >= Lm) { Hi -= Lm; Tp++; }
        if (Tp > 0) return Tp.ToString() + Hi.ToString(Fm) + Mh.ToString(Fm) + Ml.ToString(Fm) + Lo.ToString(Fm);
        if (Hi > 0) return Hi.ToString() + Mh.ToString(Fm) + Ml.ToString(Fm) + Lo.ToString(Fm);
        if (Mh > 0) return Mh.ToString() + Ml.ToString(Fm) + Lo.ToString(Fm);
        if (Ml > 0) return Ml.ToString() + Lo.ToString(Fm);
        return Lo.ToString();
    }

    static void funF(int d) { // structure of UInt64 method, high performance
        LI[] m = Co[d];
        string s = new string(d.ToString()[0], d); Write("{0}: ", d);
        for (int i = d, c = 0; c < ls; i++) {
            if (scale((uint)d, ref m[0]).Contains(s))
                Write("{0} ", i, ++c);
            for (int j = d, k = d - 1; j > 0; j = k--)
                inc(ref m[k], m[j]);
        }
        fin();
    }

    static void init() { // initializes co and Co
        for (int v = 1; v < 10; v++) {
            BI[] res = new BI[v + 1];
            long[] f = new long[v + 1], l = new long[v + 1];
            for (int j = 0; j <= v; j++) {
                if (j == v) {
                    LI[] t = new LI[v + 1];
                    for (int y = 0; y <= v; y++) set(ref t[y], (UInt64)f[y]);
                    Co.Add(t);
                }
                res[j] = f[j];
                l[0] = f[0]; f[0] = ipow(j + 1, v);
                for (int a = 0, b = 1; b <= v; a = b++) {
                    l[b] = f[b]; f[b] = f[a] - l[a];
                }
            }
            for (int z = res.Length - 2; z > 0; z -= 2) res[z] *= -1;
            co.Add(res);
        }
    }

    static DateTime st;

    static void doOne(string title, int top, Action<int> func) {
        WriteLine('\n' + title); st = DateTime.Now;
        for (int i = 2; i <= top; i++) func(i);
    }

    static void Main(string[] args)
    {
        init(); const int top = 9;
        doOne("BigInteger mostly addition:", top, funS);
        doOne("BigInteger.Pow():", top, funM);
        doOne("UInt64 structure mostly addition:", top, funF);
    }
}
Output:
BigInteger mostly addition:
2: 19 31 69 81 105 106 107 119 127 131 0.007s
3: 261 462 471 481 558 753 1036 1046 1471 1645 0.008s
4: 1168 4972 7423 7752 8431 10267 11317 11487 11549 11680 0.014s
5: 4602 5517 7539 12955 14555 20137 20379 26629 32767 35689 0.033s
6: 27257 272570 302693 323576 364509 502785 513675 537771 676657 678146 0.584s
7: 140997 490996 1184321 1259609 1409970 1783166 1886654 1977538 2457756 2714763 2.891s
8: 185423 641519 1551728 1854230 6415190 12043464 12147605 15517280 16561735 18542300 20.47s
9: 17546133 32613656 93568867 107225764 109255734 113315082 121251742 175461330 180917907 182557181 226.7s

BigInteger.Pow():
2: 19 31 69 81 105 106 107 119 127 131 0.002s
3: 261 462 471 481 558 753 1036 1046 1471 1645 0.003s
4: 1168 4972 7423 7752 8431 10267 11317 11487 11549 11680 0.008s
5: 4602 5517 7539 12955 14555 20137 20379 26629 32767 35689 0.025s
6: 27257 272570 302693 323576 364509 502785 513675 537771 676657 678146 0.450s
7: 140997 490996 1184321 1259609 1409970 1783166 1886654 1977538 2457756 2714763 2.092s
8: 185423 641519 1551728 1854230 6415190 12043464 12147605 15517280 16561735 18542300 14.80s
9: 17546133 32613656 93568867 107225764 109255734 113315082 121251742 175461330 180917907 182557181 170.8s

UInt64 structure mostly addition:
2: 19 31 69 81 105 106 107 119 127 131 0.004s
3: 261 462 471 481 558 753 1036 1046 1471 1645 0.004s
4: 1168 4972 7423 7752 8431 10267 11317 11487 11549 11680 0.006s
5: 4602 5517 7539 12955 14555 20137 20379 26629 32767 35689 0.013s
6: 27257 272570 302693 323576 364509 502785 513675 537771 676657 678146 0.188s
7: 140997 490996 1184321 1259609 1409970 1783166 1886654 1977538 2457756 2714763 1.153s
8: 185423 641519 1551728 1854230 6415190 12043464 12147605 15517280 16561735 18542300 9.783s
9: 17546133 32613656 93568867 107225764 109255734 113315082 121251742 175461330 180917907 182557181 121.3s

Results from a core i7-7700 @ 3.6Ghz. The UInt64 structure method is quicker, but it's likely because the BigInteger.ToString() implementation in C# is so sluggish. I've ported this (UInt64 structure) algorithm to C++, however it's quite a bit slower than the C++ GMP version.

Regarding the term "mostly addition", for this task there is some multiplication by a small integer (2 thru 9 in scale()), but the powers of "n" are calculated by only addition. The initial tables are calculated with multiplication and a power function (ipow()).

C

Library: GMP
#include <stdio.h>
#include <stdlib.h>
#include <string.h>
#include <gmp.h>

int main() {
    for (unsigned int d = 2; d <= 9; ++d) {
        printf("First 10 super-%u numbers:\n", d);
        char digits[16] = { 0 };
        memset(digits, '0' + d, d);
        mpz_t bignum;
        mpz_init(bignum);
        for (unsigned int count = 0, n = 1; count < 10; ++n) {
            mpz_ui_pow_ui(bignum, n, d);
            mpz_mul_ui(bignum, bignum, d);
            char* str = mpz_get_str(NULL, 10, bignum);
            if (strstr(str, digits)) {
                printf("%u ", n);
                ++count;
            }
            free(str);
        }
        mpz_clear(bignum);
        printf("\n");
    }
    return 0;
}
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 
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 
First 10 super-9 numbers:
17546133 32613656 93568867 107225764 109255734 113315082 121251742 175461330 180917907 182557181

C++

Translation of: D

There are insufficiant bits avalaible to calculate the super-5 or super-6 numbers without a biginteger library 

#include <iostream>
#include <sstream>
#include <vector>

uint64_t ipow(uint64_t base, uint64_t exp) {
    uint64_t result = 1;
    while (exp) {
        if (exp & 1) {
            result *= base;
        }
        exp >>= 1;
        base *= base;
    }
    return result;
}

int main() {
    using namespace std;

    vector<string> rd{ "22", "333", "4444", "55555", "666666", "7777777", "88888888", "999999999" };

    for (uint64_t ii = 2; ii < 5; ii++) {
        cout << "First 10 super-" << ii << " numbers:\n";
        int count = 0;

        for (uint64_t j = 3; /* empty */; j++) {
            auto k = ii * ipow(j, ii);
            auto kstr = to_string(k);
            auto needle = rd[(size_t)(ii - 2)];
            auto res = kstr.find(needle);
            if (res != string::npos) {
                count++;
                cout << j << ' ';
                if (count == 10) {
                    cout << "\n\n";
                    break;
                }
            }
        }
    }

    return 0;
}
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

Alternative using GMP

Library: GMP
Translation of: C
#include <iostream>
#include <gmpxx.h>

using big_int = mpz_class;

int main() {
    for (unsigned int d = 2; d <= 9; ++d) {
        std::cout << "First 10 super-" << d << " numbers:\n";
        std::string digits(d, '0' + d);
        big_int bignum;
        for (unsigned int count = 0, n = 1; count < 10; ++n) {
            mpz_ui_pow_ui(bignum.get_mpz_t(), n, d);
            bignum *= d;
            auto str(bignum.get_str());
            if (str.find(digits) != std::string::npos) {
                std::cout << n << ' ';
                ++count;
            }
        }
        std::cout << '\n';
    }
    return 0;
}
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 
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 
First 10 super-9 numbers:
17546133 32613656 93568867 107225764 109255734 113315082 121251742 175461330 180917907 182557181

Alternative not using GMP

Translation of: C#
#include <cstdio>
#include <sstream>
#include <chrono>

using namespace std;
using namespace chrono;

struct LI { uint64_t a, b, c, d, e; }; const uint64_t Lm = 1e18;
auto st = steady_clock::now(); LI k[10][10];

string padZ(uint64_t x, int n = 18) { string r = to_string(x);
  return string(max((int)(n - r.length()), 0), '0') + r; }

uint64_t ipow(uint64_t b, uint64_t e) { uint64_t r = 1;
  while (e) { if (e & 1) r *= b; e >>= 1; b *= b; } return r; }

uint64_t fa(uint64_t x) { // factorial
  uint64_t r = 1; while (x > 1) r *= x--; return r; }

void set(LI &d, uint64_t s) { // d = s
  d.a = s; d.b = d.c = d.d = d.e = 0; }

void inc(LI &d, LI s) { // d += s
  d.a += s.a; while (d.a >= Lm) { d.a -= Lm; d.b++; }
  d.b += s.b; while (d.b >= Lm) { d.b -= Lm; d.c++; }
  d.c += s.c; while (d.c >= Lm) { d.c -= Lm; d.d++; }
  d.d += s.d; while (d.d >= Lm) { d.d -= Lm; d.e++; }
  d.e += s.e;
}

string scale(uint32_t s, LI &x) { // multiplies x by s, converts to string
  uint64_t A = x.a * s, B = x.b * s, C = x.c * s, D = x.d * s, E = x.e * s; 
  while (A >= Lm) { A -= Lm; B++; }
  while (B >= Lm) { B -= Lm; C++; }
  while (C >= Lm) { C -= Lm; D++; }
  while (D >= Lm) { D -= Lm; E++; }
  if (E > 0) return to_string(E) + padZ(D) + padZ(C) + padZ(B) + padZ(A);
  if (D > 0) return to_string(D) + padZ(C) + padZ(B) + padZ(A);
  if (C > 0) return to_string(C) + padZ(B) + padZ(A);
  if (B > 0) return to_string(B) + padZ(A);
  return to_string(A);
}

void fun(int d) {
  auto m = k[d]; string s = string(d, '0' + d); printf("%d: ", d);
  for (int i = d, c = 0; c < 10; i++) {
    if (scale((uint32_t)d, m[0]).find(s) != string::npos) {
      printf("%d ", i); ++c; }
    for (int j = d, k = d - 1; j > 0; j = k--) inc(m[k], m[j]);
  } printf("%ss\n", to_string(duration<double>(steady_clock::now() - st).count()).substr(0,5).c_str());
}

static void init() {
  for (int v = 1; v < 10; v++) {
    uint64_t f[v + 1], l[v + 1];
    for (int j = 0; j <= v; j++) {
      if (j == v) for (int y = 0; y <= v; y++)
          set(k[v][y], v != y ? (uint64_t)f[y] : fa(v));
      l[0] = f[0]; f[0] = ipow(j + 1, v);
      for (int a = 0, b = 1; b <= v; a = b++) {
        l[b] = f[b]; f[b] = f[a] - l[a];
      }
    }
  } 
}

int main() {
  init();
  for (int i = 2; i <= 9; i++) fun(i);
}
Output:
2: 19 31 69 81 105 106 107 119 127 131 0.002s
3: 261 462 471 481 558 753 1036 1046 1471 1645 0.004s
4: 1168 4972 7423 7752 8431 10267 11317 11487 11549 11680 0.009s
5: 4602 5517 7539 12955 14555 20137 20379 26629 32767 35689 0.026s
6: 27257 272570 302693 323576 364509 502785 513675 537771 676657 678146 0.399s
7: 140997 490996 1184321 1259609 1409970 1783166 1886654 1977538 2457756 2714763 2.523s
8: 185423 641519 1551728 1854230 6415190 12043464 12147605 15517280 16561735 18542300 22.32s
9: 17546133 32613656 93568867 107225764 109255734 113315082 121251742 175461330 180917907 182557181 275.4s

Slower than GMP on the core i7-7700 @ 3.6Ghz, over twice as slow. This one is 275 seconds, vs. 102 seconds for GMP.

Clojure

Translation of: Raku

A more naïve implementation inspired by the Raku one, but without its use of parallelism; on my somewhat old and underpowered laptop, it gets through the basic task of 2 through 6 in about 6 seconds, then takes almost 40 to complete 7, and about six minutes for 8; with that growth rate, I didn't wait around for nine to complete. 

(defn super [d]
  (let [run (apply str (repeat d (str d)))]
   (filter #(clojure.string/includes? (str (* d (Math/pow % d ))) run) (range))))

(doseq [d (range 2 9)]
    (println (str d ": ") (take 10 (super d))))
Output:
2:  (19 31 69 81 105 106 107 119 127 131)
3:  (261 462 471 558 753 1046 1471 1645 1752 1848)
4:  (1168 4972 7423 7752 8431 11317 11487 11549 11680 16588)
5:  (4602 5517 7539 9469 12955 14555 20137 20379 26629 35689)
6:  (223763 302693 323576 513675 678146 1183741 1324944 1523993 1640026 1756271)
7:  (4258476 9438581 10900183 11497728 12380285 12834193 13658671 14290071 16034108 16046124)
8:  (29242698 43307276 44263715 45980752 54555936 81044125 126984952 133579963 142631696 152390251)

D

Translation of: Go
import std.bigint;
import std.conv;
import std.stdio;
import std.string;

void main() {
    auto rd = ["22", "333", "4444", "55555", "666666", "7777777", "88888888", "999999999"];
    BigInt one = 1;
    BigInt nine = 9;

    for (int ii = 2; ii <= 9; ii++) {
        writefln("First 10 super-%d numbers:", ii);
        auto count = 0;

        inner:
        for (BigInt j = 3; ; j++) {
            auto k = ii * j ^^ ii;
            auto ix = k.to!string.indexOf(rd[ii-2]);
            if (ix >= 0) {
                count++;
                write(j, ' ');
                if (count == 10) {
                    writeln();
                    writeln();
                    break inner;
                }
            }
        }
    }
}
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

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

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

F#

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

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

Fōrmulæ programs are not textual, visualization/edition of programs is done showing/manipulating structures but not text. 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 storage and transfer purposes more than visualization and edition.

Programs in Fōrmulæ are created/edited online in its website, However they run on execution servers. By default remote servers are used, but they are limited in memory and processing power, since they are intended for demonstration and casual use. A local server can be downloaded and installed, it has no limitations (it runs in your own computer). Because of that, example programs can be fully visualized and edited, but some of them will not run if they require a moderate or heavy computation/memory resources, and no local server is being used.

In this page you can see the program(s) related to this task and their results.

Go

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

Haskell

import Data.List (isInfixOf)
import Data.Char (intToDigit)

isSuperd :: (Show a, Integral a) => a -> a -> Bool
isSuperd p n =
  (replicate <*> intToDigit) (fromIntegral p) `isInfixOf` show (p * n ^ p)

findSuperd :: (Show a, Integral a) => a -> [a]
findSuperd p = filter (isSuperd p) [1 ..]

main :: IO ()
main =
  mapM_
    (putStrLn .
     ("First 10 super-" ++) .
     ((++) . show <*> ((" : " ++) . show . take 10 . findSuperd)))
    [2 .. 6]
Output:
First 10 super-2 : [19,31,69,81,105,106,107,119,127,131]
First 10 super-3 : [261,462,471,481,558,753,1036,1046,1471,1645]
First 10 super-4 : [1168,4972,7423,7752,8431,10267,11317,11487,11549,11680]
First 10 super-5 : [4602,5517,7539,12955,14555,20137,20379,26629,32767,35689]
First 10 super-6 : [27257,272570,302693,323576,364509,502785,513675,537771,676657,678146]

J

   superD=: 1 e. #~@[ E. 10 #.inv ([ * ^~)&x:

   assert    3 superD 753
   assert -. 2 superD 753


   2 3 4 5 6 ,. _ 10 {. I. 2 3 4 5 6 superD&>/i.1e6
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

Java

import java.math.BigInteger;

public class SuperDNumbers {

    public static void main(String[] args) {
        for ( int i = 2 ; i <= 9 ; i++ ) {
            superD(i, 10);
        }
    }
    
    private static final void superD(int d, int max) {
        long start = System.currentTimeMillis();
        String test = "";
        for ( int i = 0 ; i < d ; i++ ) {
            test += (""+d);
        }
        
        int n = 0;
        int i = 0;
        System.out.printf("First %d super-%d numbers: %n", max, d);
        while ( n < max ) {
            i++;
            BigInteger val = BigInteger.valueOf(d).multiply(BigInteger.valueOf(i).pow(d));
            if ( val.toString().contains(test) ) {
                n++;
                System.out.printf("%d ", i);
            }
        }
        long end = System.currentTimeMillis();
        System.out.printf("%nRun time %d ms%n%n", end-start);
        
    }

}
Output:
First 10 super-2 numbers: 
19 31 69 81 105 106 107 119 127 131 
Run time 6 ms

First 10 super-3 numbers: 
261 462 471 481 558 753 1036 1046 1471 1645 
Run time 7 ms

First 10 super-4 numbers: 
1168 4972 7423 7752 8431 10267 11317 11487 11549 11680 
Run time 13 ms

First 10 super-5 numbers: 
4602 5517 7539 12955 14555 20137 20379 26629 32767 35689 
Run time 67 ms

First 10 super-6 numbers: 
27257 272570 302693 323576 364509 502785 513675 537771 676657 678146 
Run time 546 ms

First 10 super-7 numbers: 
140997 490996 1184321 1259609 1409970 1783166 1886654 1977538 2457756 2714763 
Run time 2342 ms

First 10 super-8 numbers: 
185423 641519 1551728 1854230 6415190 12043464 12147605 15517280 16561735 18542300 
Run time 20545 ms

First 10 super-9 numbers: 
17546133 32613656 93568867 107225764 109255734 113315082 121251742 175461330 180917907 182557181 
Run time 268460 ms

jq

Works with gojq, the Go implementation of jq

The C implementation of jq does not have sufficiently accurate integer arithmetic to fulfill the task, so the output shown below is based on a run of gojq. 

# To take advantage of gojq's arbitrary-precision integer arithmetic:
def power($b): . as $in | reduce range(0;$b) as $i (1; . * $in);

# Input is $d, the number of consecutive digits, 2 <= $d <= 9
# $max is the number of superd numbers to be emitted.
def superd($number):
  . as $d
  | tostring as $s
  | ($s * $d) as $target
  | {count:0, j: 3 }
  |  while ( .count <= $number;
        .emit = null
        | if ((.j|power($d) * $d) | tostring) | index($target)
          then .count += 1
          | .emit = .j
	  else .
          end 
        | .j += 1 )
   | select(.emit).emit ;

# super-d for 2 <=d < 8
range(2; 8)
| "First 10 super-\(.) numbers:",
   superd(10)
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
First 10 super-7 numbers:
140997
490996
1184321
1259609
1409970
1783166
1886654
1977538
2457756
2714763


Julia

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)

Kotlin

Translation of: Java
import java.math.BigInteger

fun superD(d: Int, max: Int) {
    val start = System.currentTimeMillis()
    var test = ""
    for (i in 0 until d) {
        test += d
    }

    var n = 0
    var i = 0
    println("First $max super-$d numbers:")
    while (n < max) {
        i++
        val value: Any = BigInteger.valueOf(d.toLong()) * BigInteger.valueOf(i.toLong()).pow(d)
        if (value.toString().contains(test)) {
            n++
            print("$i ")
        }
    }
    val end = System.currentTimeMillis()
    println("\nRun time ${end - start} ms\n")
}

fun main() {
    for (i in 2..9) {
        superD(i, 10)
    }
}
Output:
First 10 super-2 numbers:
19 31 69 81 105 106 107 119 127 131 
Run time 31 ms

First 10 super-3 numbers:
261 462 471 481 558 753 1036 1046 1471 1645 
Run time 16 ms

First 10 super-4 numbers:
1168 4972 7423 7752 8431 10267 11317 11487 11549 11680 
Run time 25 ms

First 10 super-5 numbers:
4602 5517 7539 12955 14555 20137 20379 26629 32767 35689 
Run time 290 ms

First 10 super-6 numbers:
27257 272570 302693 323576 364509 502785 513675 537771 676657 678146 
Run time 1203 ms

First 10 super-7 numbers:
140997 490996 1184321 1259609 1409970 1783166 1886654 1977538 2457756 2714763 
Run time 3876 ms

First 10 super-8 numbers:
185423 641519 1551728 1854230 6415190 12043464 12147605 15517280 16561735 18542300 
Run time 29817 ms

First 10 super-9 numbers:
17546133 32613656 93568867 107225764 109255734 113315082 121251742 175461330 180917907 182557181 
Run time 392181 ms

Lua

Lua - using doubles

Lua's default numeric type is IEEE-754 doubles, which are insufficient beyond d=5, but for reference: 

for d = 2, 5 do
  local n, found = 0, {}
  local dds = string.rep(d, d)
  while #found < 10 do
    local dnd = string.format("%15.f", d * n ^ d)
    if string.find(dnd, dds) then found[#found+1] = n end
    n = n + 1
  end
  print("super-" .. d .. ":  " .. table.concat(found,", "))
end
Output:
super-2:  19, 31, 69, 81, 105, 106, 107, 119, 127, 131
super-3:  261, 462, 471, 481, 558, 753, 1036, 1046, 1471, 1645
super-4:  1168, 4972, 7423, 7752, 8431, 10267, 11317, 11487, 11549, 11680
super-5:  4602, 5517, 7539, 9469, 12955, 14555, 20137, 20379, 26629, 35689

Lua - using lbc

Using the lbc library to provide support for arbitrary-precision integers, which are sufficient for both task and extra-credit: 

bc = require("bc")
for i = 2, 9 do
  local d, n, found = bc.new(i), bc.new(0), {}
  local dds = string.rep(d:tostring(), d:tonumber())
  while #found < 10 do
    local dnd = (d * n ^ d):tostring()
    if string.find(dnd, dds) then found[#found+1] = n:tostring() end
    n = n + 1
  end
  print("super-" .. d:tostring() .. ":  " .. table.concat(found,", "))
end
Output:
super-2:  19, 31, 69, 81, 105, 106, 107, 119, 127, 131
super-3:  261, 462, 471, 481, 558, 753, 1036, 1046, 1471, 1645
super-4:  1168, 4972, 7423, 7752, 8431, 10267, 11317, 11487, 11549, 11680
super-5:  4602, 5517, 7539, 12955, 14555, 20137, 20379, 26629, 32767, 35689
super-6:  27257, 272570, 302693, 323576, 364509, 502785, 513675, 537771, 676657, 678146
super-7:  140997, 490996, 1184321, 1259609, 1409970, 1783166, 1886654, 1977538, 2457756, 2714763
super-8:  185423, 641519, 1551728, 1854230, 6415190, 12043464, 12147605, 15517280, 16561735, 18542300
super-9:  17546133, 32613656, 93568867, 107225764, 109255734, 113315082, 121251742, 175461330, 180917907, 182557181

Mathematica / Wolfram Language

ClearAll[SuperD]
SuperD[d_, m_] := Module[{n, res, num},
  res = {};
  n = 1;
  While[Length[res] < m,
   num = IntegerDigits[d n^d];
   If[MatchQ[num, {___, Repeated[d, {d}], ___}],
    AppendTo[res, n]
    ];
   n++;
   ];
  res
  ]
Scan[Print[SuperD[#, 10]] &, Range[2, 6]]
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}

Nim

Library: bignum

Using only the standard library, we would be limited to "d = 2, 3, 4". So, here is the program using the third-party library "bignum". 

import sequtils, strutils, times
import bignum

iterator superDNumbers(d, maxCount: Positive): Natural =
  var count = 0
  var n = 2
  let e = culong(d)   # Bignum ^ requires a culong as exponent.
  let pattern = repeat(chr(d + ord('0')), d)
  while count != maxCount:
    if pattern in $(d * n ^ e):
      yield n
      inc count
    inc n, 1

let t0 = getTime()
for d in 2..9:
  echo "First 10 super-$# numbers:".format(d)
  echo toSeq(superDNumbers(d, 10)).join(" ")
echo "Time: ", getTime() - t0
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
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
First 10 super-9 numbers:
17546133 32613656 93568867 107225764 109255734 113315082 121251742 175461330 180917907 182557181
Time: 1 minute, 43 seconds, 647 milliseconds, 920 microseconds, and 938 nanoseconds

Pascal

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

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

Phix

Translation of: Go
with javascript_semantics
include mpfr.e
 
procedure main()
    atom t0 = time()
    mpz k = mpz_init()
    for i=2 to iff(platform()=JS?7:9) 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>

Since its actually about 3 times faster under pwa/p2js, I let that one run, just the once, keeping the above capped at 7 for ~11s: 

...
First 10 super-8 numbers:
185423 641519 1551728 1854230 6415190 12043464 12147605 15517280 16561735 18542300 
found in 1 minute and 22s

First 10 super-9 numbers:
17546133 32613656 93568867 107225764 109255734 113315082 121251742 175461330 180917907 182557181 
found in 16 minutes and 6s

Python

Procedural

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, 9):
        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

Functional

'''Super-d numbers'''

from itertools import count, islice
from functools import reduce


# ------------------------ SUPER-D -------------------------

# super_d :: Int -> Either String [Int]
def super_d(d):
    '''Either a message, if d is out of range, or
       an infinite series of super_d numbers for d.
    '''
    if isinstance(d, int) and 1 < d < 10:
        ds = d * str(d)

        def p(x):
            return ds in str(d * x ** d)

        return Right(filter(p, count(2)))
    else:
        return Left(
            'Super-d is defined only for integers drawn from {2..9}'
        )


# ------------------------- TESTS --------------------------
# main :: IO ()
def main():
    '''Attempted sampling of first 10 values for d <- [1..6],
       where d = 1 is out of range.
    '''
    for v in map(
        lambda x: either(
            append(str(x) + ' :: ')
        )(
            compose(
                append('First 10 super-' + str(x) + ': '),
                showList
            )
        )(
            bindLR(
                super_d(x)
            )(compose(Right, take(10)))
        ),
        enumFromTo(1)(6)
    ): print(v)


# ------------------------ GENERIC -------------------------

# Left :: a -> Either a b
def Left(x):
    '''Constructor for an empty Either (option type) value
       with an associated string.
    '''
    return {'type': 'Either', 'Right': None, 'Left': x}


# Right :: b -> Either a b
def Right(x):
    '''Constructor for a populated Either (option type) value'''
    return {'type': 'Either', 'Left': None, 'Right': x}


# append (++) :: [a] -> [a] -> [a]
# append (++) :: String -> String -> String
def append(xs):
    '''A list or string formed by
       the concatenation of two others.
    '''
    def go(ys):
        return xs + ys
    return go


# bindLR (>>=) :: Either a -> (a -> Either b) -> Either b
def bindLR(m):
    '''Either monad injection operator.
       Two computations sequentially composed,
       with any value produced by the first
       passed as an argument to the second.
    '''
    def go(mf):
        return (
            mf(m.get('Right')) if None is m.get('Left') else m
        )
    return go


# compose :: ((a -> a), ...) -> (a -> a)
def compose(*fs):
    '''Composition, from right to left,
       of a series of functions.
    '''
    def go(f, g):
        def fg(x):
            return f(g(x))
        return fg
    return reduce(go, fs, lambda x: x)


# either :: (a -> c) -> (b -> c) -> Either a b -> c
def either(fl):
    '''The application of fl to e if e is a Left value,
       or the application of fr to e if e is a Right value.
    '''
    return lambda fr: lambda e: fl(e['Left']) if (
        None is e['Right']
    ) else fr(e['Right'])


# enumFromTo :: Int -> Int -> [Int]
def enumFromTo(m):
    '''Enumeration of integer values [m..n]
    '''
    return lambda n: range(m, 1 + n)


# showList :: [a] -> String
def showList(xs):
    '''Stringification of a list.'''
    return '[' + ','.join(str(x) for x in xs) + ']'


# take :: Int -> [a] -> [a]
# take :: Int -> String -> String
def take(n):
    '''The prefix of xs of length n,
       or xs itself if n > length xs.
    '''
    def go(xs):
        return islice(xs, n)
    return go


# MAIN ---
if __name__ == '__main__':
    main()
Output:
1 :: Super-d is defined only for integers drawn from {2..9}
First 10 super-2: [19,31,69,81,105,106,107,119,127,131]
First 10 super-3: [261,462,471,481,558,753,1036,1046,1471,1645]
First 10 super-4: [1168,4972,7423,7752,8431,10267,11317,11487,11549,11680]
First 10 super-5: [4602,5517,7539,12955,14555,20137,20379,26629,32767,35689]
First 10 super-6: [27257,272570,302693,323576,364509,502785,513675,537771,676657,678146]


R

Library: Rmpfr

Library is necessary to go beyond super-4 numbers. Indeed Rmpfr allows to augment precision for floating point numbers. I didn't go for extra task, because it took some minutes for super-6 numbers. 

library(Rmpfr)
options(scipen = 999)

find_super_d_number <- function(d, N = 10){
  
  super_number <- c(NA)
  n = 0
  n_found = 0
  
  while(length(super_number) < N){
    
    n = n + 1
    test = d * mpfr(n, precBits = 200) ** d   #Here I augment precision
    test_formatted = .mpfr2str(test)$str      #and I extract the string from S4 class object
    
    iterable = strsplit(test_formatted, "")[[1]]
    
    if (length(iterable) < d) next
    
    for(i in d:length(iterable)){
      if (iterable[i] != d) next 
      equalities = 0
      
      for(j in 1:d) {
        if (i == j) break 
        
        if(iterable[i] == iterable[i-j])
          equalities = equalities + 1
        else break
      }
      
      if (equalities >= (d-1)) {
        n_found = n_found + 1
        super_number[n_found] = n
        break
      }
    }    
  }
  
  message(paste0("First ", N, " super-", d, " numbers:"))
  print((super_number))
  
  return(super_number)
}

for(d in 2:6){find_super_d_number(d, N = 10)}
Output:
First 10 super-2 numbers:
  [1]  19  31  69  81 105 106 107 119 127 131
First 10 super-3 numbers:
  [1]  261  462  471  481  558  753 1036 1046 1471 1645
First 10 super-4 numbers:
  [1]  1168  4972  7423  7752  8431 10267 11317 11487 11549 11680
First 10 super-5 numbers:
  [1]  4602  5517  7539 12955 14555 20137 20379 26629 32767 35689
First 10 super-6 numbers:
  [1]  27257 272570 302693 323576 364509 502785 513675 537771 676657 678146

Raku

(formerly Perl 6)

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.

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  "   "  "   "   "     "      "  */
                                                 /* [↓]   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

Ruby

(2..8).each do |d|
  rep = d.to_s * d
  print "#{d}: "
  puts (2..).lazy.select{|n| (d * n**d).to_s.include?(rep) }.first(10).join(", ")
end
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

The first 10 super_9 numbers (not shown) take about 540 seconds.

Rust

// [dependencies]
// rug = "1.9"

fn print_super_d_numbers(d: u32, limit: u32) {
    use rug::Assign;
    use rug::Integer;

    println!("First {} super-{} numbers:", limit, d);
    let digits = d.to_string().repeat(d as usize);
    let mut count = 0;
    let mut n = 1;
    let mut s = Integer::new();
    while count < limit {
        s.assign(Integer::u_pow_u(n, d));
        s *= d;
        if s.to_string().contains(&digits) {
            print!("{} ", n);
            count += 1;
        }
        n += 1;
    }
    println!();
}

fn main() {
    for d in 2..=9 {
        print_super_d_numbers(d, 10);
    }
}
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 
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 
First 10 super-9 numbers:
17546133 32613656 93568867 107225764 109255734 113315082 121251742 175461330 180917907 182557181

Sidef

func super_d(d) {
    var D = Str(d)*d
    1..Inf -> lazy.grep {|n| Str(d * n**d).contains(D) }
}

for d in (2..8) {
    say ("#{d}: ", super_d(d).first(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]

Swift

Library: AttaSwift BigInt
import BigInt
import Foundation

let rd = ["22", "333", "4444", "55555", "666666", "7777777", "88888888", "999999999"]

for d in 2...9 {
  print("First 10 super-\(d) numbers:")

  var count = 0
  var n = BigInt(3)
  var k = BigInt(0)

  while true {
    k = n.power(d)
    k *= BigInt(d)

    if let _ = String(k).range(of: rd[d - 2]) {
      count += 1

      print(n, terminator: " ")
      fflush(stdout)

      guard count < 10 else {
        break
      }
    }

    n += 1
  }

  print()
  print()
}
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 

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 

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

Visual Basic .NET

Translation of: D
Imports System.Numerics

Module Module1

    Sub Main()
        Dim rd = {"22", "333", "4444", "55555", "666666", "7777777", "88888888", "999999999"}
        Dim one As BigInteger = 1
        Dim nine As BigInteger = 9

        For ii = 2 To 9
            Console.WriteLine("First 10 super-{0} numbers:", ii)
            Dim count = 0

            Dim j As BigInteger = 3
            While True
                Dim k = ii * BigInteger.Pow(j, ii)
                Dim ix = k.ToString.IndexOf(rd(ii - 2))
                If ix >= 0 Then
                    count += 1
                    Console.Write("{0} ", j)
                    If count = 10 Then
                        Console.WriteLine()
                        Console.WriteLine()
                        Exit While
                    End If
                End If

                j += 1
            End While
        Next
    End Sub

End Module
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

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

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

Wren

Translation of: Go

CLI (BigInt)

Library: Wren-big
Library: Wren-fmt

Managed to get up to 8 but too slow for 9. 

import "/big" for BigInt
import "/fmt" for Fmt

var start = System.clock
var rd = ["22", "333", "4444", "55555", "666666", "7777777", "88888888"]
for (i in 2..8) {
    Fmt.print("First 10 super-$d numbers:", i)
    var count = 0
    var j = BigInt.three
    while (true) {
        var k = j.pow(i) * i
        var ix = k.toString.indexOf(rd[i-2])
        if (ix >= 0) {
            count = count + 1
            Fmt.write("$i ", j)
            if (count == 10) {
                Fmt.print("\nfound in $f seconds\n", System.clock - start)
                break
            }
        }
        j = j.inc
    }
}
Output:
First 10 super-2 numbers:
19 31 69 81 105 106 107 119 127 131 
found in 0.000500 seconds

First 10 super-3 numbers:
261 462 471 481 558 753 1036 1046 1471 1645 
found in 0.007636 seconds

First 10 super-4 numbers:
1168 4972 7423 7752 8431 10267 11317 11487 11549 11680 
found in 0.070462 seconds

First 10 super-5 numbers:
4602 5517 7539 12955 14555 20137 20379 26629 32767 35689 
found in 0.388554 seconds

First 10 super-6 numbers:
27257 272570 302693 323576 364509 502785 513675 537771 676657 678146 
found in 10.044345 seconds

First 10 super-7 numbers:
140997 490996 1184321 1259609 1409970 1783166 1886654 1977538 2457756 2714763 
found in 62.021899 seconds

First 10 super-8 numbers:
185423 641519 1551728 1854230 6415190 12043464 12147605 15517280 16561735 18542300 
found in 434.536825 seconds

Embedded (GMP)

Library: Wren-gmp

Much sprightlier with 8 now being reached in 11.7 seconds and 9 in 126.5 seconds. 

/* super-d_numbers_gmp.wren */

import "./gmp" for Mpz
import "./fmt" for Fmt
 
var start = System.clock
var rd = ["22", "333", "4444", "55555", "666666", "7777777", "88888888", "999999999"]
for (i in 2..9) {
    Fmt.print("First 10 super-$d numbers:", i)
    var count = 0
    var j = Mpz.three
    var k = Mpz.new()
    while (true) {
        k.pow(j, i).mul(i)
        var ix = k.toString.indexOf(rd[i-2])
        if (ix >= 0) {
            count = count + 1
            Fmt.write("$i ", j)
            if (count == 10) {
                Fmt.print("\nfound in $f seconds\n", System.clock - start)
                break
            }
        }
        j.inc
    }
}
Output:
First 10 super-2 numbers:
19 31 69 81 105 106 107 119 127 131 
found in 0.000222 seconds

First 10 super-3 numbers:
261 462 471 481 558 753 1036 1046 1471 1645 
found in 0.001222 seconds

First 10 super-4 numbers:
1168 4972 7423 7752 8431 10267 11317 11487 11549 11680 
found in 0.007386 seconds

First 10 super-5 numbers:
4602 5517 7539 12955 14555 20137 20379 26629 32767 35689 
found in 0.024863 seconds

First 10 super-6 numbers:
27257 272570 302693 323576 364509 502785 513675 537771 676657 678146 
found in 0.339837 seconds

First 10 super-7 numbers:
140997 490996 1184321 1259609 1409970 1783166 1886654 1977538 2457756 2714763 
found in 1.684412 seconds

First 10 super-8 numbers:
185423 641519 1551728 1854230 6415190 12043464 12147605 15517280 16561735 18542300 
found in 11.695570 seconds

First 10 super-9 numbers:
17546133 32613656 93568867 107225764 109255734 113315082 121251742 175461330 180917907 182557181 
found in 126.454911 seconds

zkl

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
Retrieved from "https://rosettacode.org/wiki/Super-d_numbers?oldid=328553"