Smarandache-Wellin primes

Smarandache-Wellin primes
You are encouraged to solve this task according to the task description, using any language you may know.

Definitions

A Smarandache-Wellin number (S-W number for short) is an integer that in a given base is the concatenation of the first n prime numbers written in that base. A base of 10 will be assumed for this task.

A Derived S-W number (not an 'official' term) is an integer formed from a S-W number by working out the number of times each of the digits 0 to 9 occurs in that number, concatenating those frequencies in the same order (i.e. frequency of '0' first, frequency of '1' second etc) and removing any leading zeros.

Examples

'23571113' is the sixth S-W number formed by concatenating the first 6 primes: 2, 3, 5, 7, 11 and 13.

The corresponding Derived S-W number is '312010100' because '1' occurs 3 times, '3' occurs twice and '2', '5' and '7' all occur once.

Task

• Find and show the first three S-W numbers which are prime.

• Find and show the first three Derived S-W numbers which are prime.

Stretch (requires 'big integer' support)

Find and show the index in the sequence (starting from 1), the total number of digits and the last prime used to form the fourth, fifth, sixth, seventh and (optionally) the eighth S-W numbers which are prime or probably prime with reasonable certainty. It is unknown whether there are any more but, if you fancy searching for one, good luck! You can start from an index of 22,077.

Also find and show up to the first twenty Derived S-W numbers which are prime and their index in the sequence.

References

• Wikipedia: Smarandache-Wellin number
• OEIS:A019518 - Smarandache-Wellin numbers
• OEIS:A069151 - Smarandache-Wellin primes

C

Translation of: Wren

Library: GMP

#include <stdio.h>
#include <stdlib.h>
#include <stdbool.h>
#include <string.h>
#include <stdint.h>
#include <gmp.h>

bool *sieve(int limit) {
    int i, p;
    limit++;
    // True denotes composite, false denotes prime.
    bool *c = calloc(limit, sizeof(bool)); // all false by default
    c[0] = true;
    c[1] = true;
    for (i = 4; i < limit; i += 2) c[i] = true;
    p = 3; // Start from 3.
    while (true) {
        int p2 = p * p;
        if (p2 >= limit) break;
        for (i = p2; i < limit; i += 2 * p) c[i] = true;
        while (true) {
            p += 2;
            if (!c[p]) break;
        }
    }
    return c;
}

const char *ord(int count) {
    return (count == 1) ? "st" :
           (count == 2) ? "nd" :
           (count == 3) ? "rd" : "th";
}   

int main() {
    bool *c = sieve(12000);
    char sw[6000] = "";
    char tmp[20];
    int count = 0, p = 1, ix = 0, i;
    mpz_t n;
    mpz_init(n);
    printf("The known Smarandache-Wellin primes are:\n");
    while (count < 8) {
        while (c[++p]);
        sprintf(tmp, "%d", p);
        strcat(sw, tmp);
        mpz_set_str(n, sw, 10);
        if (mpz_probab_prime_p(n, 15) > 0) {
            count++;
            size_t le = strlen(sw);
            char sws[44];
            if (le < 5) {
                strcpy(sws, sw);
            } else {
                strncpy(sws, sw, 20);
                strcpy(sws + 20, "...");
                strncpy(sws + 23, sw + le - 20, 21);
            }
            printf("%2d%s: index %4d  digits %4ld  last prime %5d -> %s\n", count, ord(count), ix+1, le, p, sws);
        }
        ix++;
    }

    printf("\nThe first 20 Derived Smarandache-Wellin primes are:\n");
    int freqs[10] = {0};
    count = 0;
    p = 1;
    ix = 0;
    while (count < 20) {
        while (c[++p]);
        sprintf(tmp, "%d", p);
        for (i = 0; i < strlen(tmp); ++i) freqs[tmp[i]-48]++;
        char dsw[40] = "";
        for (i = 0; i < 10; ++i) {
            sprintf(tmp, "%d", freqs[i]);
            strcat(dsw, tmp);
        }
        int idx = -1;
        for (i = 0; i < strlen(dsw); ++i) {
            if (dsw[i] != '0') {
                idx = i;
                break;
            }
        }
        mpz_set_str(n, dsw + idx, 10);
        if (mpz_probab_prime_p(n, 15) > 0) {
            count++;                                                      
            printf("%2d%s: index %4d  prime %s\n", count, ord(count), ix+1, dsw + idx);
        }
        ix++; 
    }
    free(c);
    return 0;
}

Output:

The known Smarandache-Wellin primes are:
 1st: index    1  digits    1  last prime     2 -> 2
 2nd: index    2  digits    2  last prime     3 -> 23
 3rd: index    4  digits    4  last prime     7 -> 2357
 4th: index  128  digits  355  last prime   719 -> 23571113171923293137...73677683691701709719
 5th: index  174  digits  499  last prime  1033 -> 23571113171923293137...10131019102110311033
 6th: index  342  digits 1171  last prime  2297 -> 23571113171923293137...22732281228722932297
 7th: index  435  digits 1543  last prime  3037 -> 23571113171923293137...30013011301930233037
 8th: index 1429  digits 5719  last prime 11927 -> 23571113171923293137...11903119091192311927

The first 20 Derived Smarandache-Wellin primes are:
 1st: index   32  prime 4194123321127
 2nd: index   72  prime 547233879626521
 3rd: index   73  prime 547233979727521
 4th: index  134  prime 13672766322929571043
 5th: index  225  prime 3916856106393739943689
 6th: index  303  prime 462696313560586013558131
 7th: index  309  prime 532727113760586013758133
 8th: index  363  prime 6430314317473636515467149
 9th: index  462  prime 8734722823685889120488197
10th: index  490  prime 9035923128899919621189209
11th: index  495  prime 9036023329699969621389211
12th: index  522  prime 9337023533410210710923191219
13th: index  538  prime 94374237357103109113243102223
14th: index  624  prime 117416265406198131121272110263
15th: index  721  prime 141459282456260193137317129313
16th: index  738  prime 144466284461264224139325131317
17th: index  790  prime 156483290479273277162351153339
18th: index  852  prime 164518312512286294233375158359
19th: index 1087  prime 208614364610327343341589284471
20th: index 1188  prime 229667386663354357356628334581

C++

Library: GMP

Library: Primesieve

#include <iomanip>
#include <iostream>
#include <string>

#include <primesieve.hpp>

#include <gmpxx.h>

using big_int = mpz_class;

class sw_number_generator {
public:
    sw_number_generator() { next(); }
    void next();
    const std::string& number() const { return number_; }
    uint64_t prime() const { return prime_; }
    int index() const { return index_; }
    std::string derived_number() const;

private:
    primesieve::iterator pi_;
    std::string number_;
    uint64_t prime_;
    int index_ = 0;
};

void sw_number_generator::next() {
    ++index_;
    prime_ = pi_.next_prime();
    number_ += std::to_string(prime_);
}

std::string sw_number_generator::derived_number() const {
    int count[10] = {};
    for (char c : number_)
        ++count[c - '0'];
    std::string str;
    for (int i = 0; i < 10; ++i) {
        if (!str.empty() || count[i] != 0)
            str += std::to_string(count[i]);
    }
    return str;
}

bool is_probably_prime(const big_int& n) {
    return mpz_probab_prime_p(n.get_mpz_t(), 25) != 0;
}

std::string abbreviate(std::string str, size_t max_digits) {
    size_t len = str.size();
    if (len > max_digits)
        str.replace(max_digits / 2, len - max_digits, "...");
    return str;
}

void print_sw_primes() {
    std::cout
        << "Known Smarandache-Wellin primes:\n\n"
        << "  |Index|Digits|Last prime|         Smarandache-Wellin prime\n"
        << "----------------------------------------------------------------------\n";
    sw_number_generator swg;
    for (int n = 1; n <= 8; swg.next()) {
        std::string num = swg.number();
        if (is_probably_prime(big_int(num))) {
            std::cout << std::setw(2) << n << "|"
                      << std::setw(5) << swg.index() << "|"
                      << std::setw(6) << num.size() << "|"
                      << std::setw(10) << swg.prime() << "|"
                      << std::setw(43) << abbreviate(num, 40) << '\n';
            ++n;
        }
    }
}

void print_derived_sw_primes(int count) {
    std::cout << "First " << count << " Derived Smarandache-Wellin primes:\n\n"
              << "  |Index|Derived Smarandache-Wellin prime\n"
              << "-----------------------------------------\n";
    sw_number_generator swg;
    for (int n = 1; n <= count; swg.next()) {
        std::string num = swg.derived_number();
        if (is_probably_prime(big_int(num))) {
            std::cout << std::setw(2) << n << "|"
                      << std::setw(5) << swg.index() << "|"
                      << std::setw(32) << num << '\n';
            ++n;
        }
    }
}

int main() {
    print_sw_primes();
    std::cout << '\n';
    print_derived_sw_primes(20);
}

Output:

Known Smarandache-Wellin primes:

  |Index|Digits|Last prime|         Smarandache-Wellin prime
----------------------------------------------------------------------
 1|    1|     1|         2|                                          2
 2|    2|     2|         3|                                         23
 3|    4|     4|         7|                                       2357
 4|  128|   355|       719|23571113171923293137...73677683691701709719
 5|  174|   499|      1033|23571113171923293137...10131019102110311033
 6|  342|  1171|      2297|23571113171923293137...22732281228722932297
 7|  435|  1543|      3037|23571113171923293137...30013011301930233037
 8| 1429|  5719|     11927|23571113171923293137...11903119091192311927

First 20 Derived Smarandache-Wellin primes:

  |Index|Derived Smarandache-Wellin prime
-----------------------------------------
 1|   32|                   4194123321127
 2|   72|                 547233879626521
 3|   73|                 547233979727521
 4|  134|            13672766322929571043
 5|  225|          3916856106393739943689
 6|  303|        462696313560586013558131
 7|  309|        532727113760586013758133
 8|  363|       6430314317473636515467149
 9|  462|       8734722823685889120488197
10|  490|       9035923128899919621189209
11|  495|       9036023329699969621389211
12|  522|    9337023533410210710923191219
13|  538|   94374237357103109113243102223
14|  624|  117416265406198131121272110263
15|  721|  141459282456260193137317129313
16|  738|  144466284461264224139325131317
17|  790|  156483290479273277162351153339
18|  852|  164518312512286294233375158359
19| 1087|  208614364610327343341589284471
20| 1188|  229667386663354357356628334581

EasyLang

fastfunc isprim num .
   i = 2
   while i <= sqrt num
      if num mod i = 0
         return 0
      .
      i += 1
   .
   return 1
.
func nextprim num .
   while isprim num = 0
      num += 1
   .
   return num
.
len digs[] 10
prim = 1
while len prsw[] < 3 or len prdsw[] < 3
   prim = nextprim (prim + 1)
   h = prim
   h[] = [ ]
   while h > 0
      d = h mod 10
      digs[d + 1] += 1
      h[] &= d
      h = h div 10
   .
   for i = len h[] downto 1
      sw = sw * 10 + h[i]
   .
   if isprim sw = 1
      prsw[] &= sw
   .
   dsw = 0
   for i to 10
      if digs[i] = 0
         h = 10
      else
         h = 1 + floor log10 digs[i]
         h = pow 10 h
      .
      dsw = dsw * h + digs[i]
   .
   if isprim dsw = 1
      prdsw[] &= dsw
   .
.
print prsw[]
print prdsw[]

Output:

[ 2 23 2357 ]
[ 4194123321127 547233879626521 547233979727521 ]

F#

// Smarandache-Wellin primes. Nigel Galloway: April 3rd., 2023
let izP(g,_,_,_)=let mutable g=System.Numerics.BigInteger.Parse g
                 Open.Numeric.Primes.MillerRabin.IsProbablePrime &g
let fN g="0123456789"+g|>Seq.countBy id|>Seq.map(fun(_,g)->string(g-1))|>Seq.fold((+))""
let sw()=primes32()|>Seq.scan(fun(n,i,g,e)l->let a=string l in (n+a,l,g+1,e+(String.length a)))("",0,0,0)|>Seq.skip 1
sw()|>Seq.filter izP|>Seq.take 3|>Seq.iter(fun(i,_,_,_)->printf "%s " i); printfn ""
sw()|>Seq.filter izP|>Seq.take 7|>Seq.iter(fun(_,n,g,l)->printfn "index %4d: digits %4d last prime %d" g l n); printfn ""
sw()|>Seq.map(fun(i,g,e,l)->(fN i,g,e,l))|>Seq.filter izP|>Seq.take 20|>Seq.iter(fun(n,_,g,_)->printfn $"index %4d{g}: prime %s{n}")

Output:

2 23 2357
index    1: digits    1 last prime 2
index    2: digits    2 last prime 3
index    4: digits    4 last prime 7
index  128: digits  355 last prime 719
index  174: digits  499 last prime 1033
index  342: digits 1171 last prime 2297
index  435: digits 1543 last prime 3037
index   32: prime 4194123321127
index   72: prime 547233879626521
index   73: prime 547233979727521
index  134: prime 13672766322929571043
index  225: prime 3916856106393739943689
index  303: prime 462696313560586013558131
index  309: prime 532727113760586013758133
index  363: prime 6430314317473636515467149
index  462: prime 8734722823685889120488197
index  490: prime 9035923128899919621189209
index  495: prime 9036023329699969621389211
index  522: prime 9337023533410210710923191219
index  538: prime 94374237357103109113243102223
index  624: prime 117416265406198131121272110263
index  721: prime 141459282456260193137317129313
index  738: prime 144466284461264224139325131317
index  790: prime 156483290479273277162351153339
index  852: prime 164518312512286294233375158359
index 1087: prime 208614364610327343341589284471
index 1188: prime 229667386663354357356628334581

Go

Translation of: Wren

Library: Go-rcu

Library: GMP(Go wrapper)

We need to use the above GMP wrapper to match Wren's time of about 35.5 seconds as Go's native big.Int type is far slower.

package main

import (
    "fmt"
    big "github.com/ncw/gmp"
    "rcu"
    "strconv"
    "strings"
)

func ord(count int) string {
    if count == 1 {
        return "st"
    }
    if count == 2 {
        return "nd"
    }
    if count == 3 {
        return "rd"
    }
    return "th"
}

func main() {
    primes := rcu.Primes(12000)
    sw := ""
    count := 0
    i := 0
    n := new(big.Int)
    fmt.Println("The known Smarandache-Wellin primes are:")
    for count < 8 {
        sw += strconv.Itoa(primes[i])
        n.SetString(sw, 10)
        if n.ProbablyPrime(15) {
            count++
            sws := sw
            le := len(sws)
            if le > 4 {
                sws = sws[0:20] + "..." + sws[le-20:le]
            }
            fmt.Printf("%d%s: index %4d  digits %4d  last prime %5d -> %s\n", count, ord(count), i+1, len(sw), primes[i], sws)
        }
        i++
    }

    fmt.Println("\nThe first 20 Derived Smarandache-Wellin primes are:")
    freqs := make([]int, 10)
    count = 0
    i = 0
    for count < 20 {
        p := strconv.Itoa(primes[i])
        for _, d := range p {
            n, _ := strconv.Atoi(string(d))
            freqs[n]++
        }
        dsw := ""
        for _, freq := range freqs {
            dsw += strconv.Itoa(freq)
        }
        dsw = strings.TrimLeft(dsw, "0")
        n.SetString(dsw, 10)
        if n.ProbablyPrime(15) {
            count++
            fmt.Printf("%2d%s: index %4d  prime %v\n", count, ord(count), i+1, n)
        }
        i++
    }
}

Output:

The known Smarandache-Wellin primes are:
1st: index    1  digits    1  last prime     2 -> 2
2nd: index    2  digits    2  last prime     3 -> 23
3rd: index    4  digits    4  last prime     7 -> 2357
4th: index  128  digits  355  last prime   719 -> 23571113171923293137...73677683691701709719
5th: index  174  digits  499  last prime  1033 -> 23571113171923293137...10131019102110311033
6th: index  342  digits 1171  last prime  2297 -> 23571113171923293137...22732281228722932297
7th: index  435  digits 1543  last prime  3037 -> 23571113171923293137...30013011301930233037
8th: index 1429  digits 5719  last prime 11927 -> 23571113171923293137...11903119091192311927

The first 20 Derived Smarandache-Wellin primes are:
 1st: index   32  prime 4194123321127
 2nd: index   72  prime 547233879626521
 3rd: index   73  prime 547233979727521
 4th: index  134  prime 13672766322929571043
 5th: index  225  prime 3916856106393739943689
 6th: index  303  prime 462696313560586013558131
 7th: index  309  prime 532727113760586013758133
 8th: index  363  prime 6430314317473636515467149
 9th: index  462  prime 8734722823685889120488197
10th: index  490  prime 9035923128899919621189209
11th: index  495  prime 9036023329699969621389211
12th: index  522  prime 9337023533410210710923191219
13th: index  538  prime 94374237357103109113243102223
14th: index  624  prime 117416265406198131121272110263
15th: index  721  prime 141459282456260193137317129313
16th: index  738  prime 144466284461264224139325131317
17th: index  790  prime 156483290479273277162351153339
18th: index  852  prime 164518312512286294233375158359
19th: index 1087  prime 208614364610327343341589284471
20th: index 1188  prime 229667386663354357356628334581

J

   derive=. [: <:@#/.~ i.@10 , ]
   digits=. ;@:(<@("."0@":)"0)
      
   (#~ 1&p:) 10&#.@digits\ p: i. 10
2 23 2357

   (#~ 1 p: {:"1) (# , 10x&#.@digits@derive@digits)\ p: i. 1200
  32                  4194123321127
  72                547233879626521
  73                547233979727521
 134           13672766322929571043
 225         3916856106393739943689
 303       462696313560586013558131
 309       532727113760586013758133
 363      6430314317473636515467149
 462      8734722823685889120488197
 490      9035923128899919621189209
 495      9036023329699969621389211
 522   9337023533410210710923191219
 538  94374237357103109113243102223
 624 117416265406198131121272110263
 721 141459282456260193137317129313
 738 144466284461264224139325131317
 790 156483290479273277162351153339
 852 164518312512286294233375158359
1087 208614364610327343341589284471
1188 229667386663354357356628334581

Java

import java.math.BigInteger;
import java.util.Arrays;
import java.util.List;
import java.util.stream.Collectors;
import java.util.stream.Stream;

public final class SmarandacheWellinPrimes {

	public static void main(String[] args) {
		primes = listPrimesUpTo(12_000);
		smarandacheWellinPrimes();
		System.out.println();
		derivedSmarandacheWellinPrimes();
	}
	
	private static void smarandacheWellinPrimes() {
		int count = 0;
		int index = 1;
		int number = 2;
		String numberString = "2";
		System.out.println("The first eight Smarandache-Wellin primes are:");
		
		while ( count < 8 ) {
			if ( new BigInteger(numberString).isProbablePrime(CERTAINTY_LEVEL) ) {
				count += 1;
				System.out.println(String.format("%2d%s%-4d%s%-5d%s%d", count, ":  index = ", index,
					"  last prime = ", number, "  number of digits = ", numberString.length()));
			}			
			
			number = primes.get(index);
			numberString += number;
			index += 1;
		}
	}
	
	private static void derivedSmarandacheWellinPrimes() {		
 		int count = 0;
		int index = 0;
		int[] digitFrequencies = new int[10];
		System.out.println("The first 20 derived Smarandache-Wellin numbers which are prime:");
		
		while ( count < 20 ) {
			String prime = String.valueOf(primes.get(index));
			for ( char ch : prime.toCharArray() ) {
				digitFrequencies[ch - '0'] += 1;
			}
			
			index += 1;
			String joined = Arrays.stream(digitFrequencies).mapToObj(String::valueOf).collect(Collectors.joining(""));
			if ( new BigInteger(joined).isProbablePrime(CERTAINTY_LEVEL) ) {
				count += 1;
				System.out.println(String.format("%2d%s%-4d%s%s", count, ":  index = ", index, "  prime = ", joined));
			}
		}
	}
	
	private static List<Integer> listPrimesUpTo(int limit) {
		final int halfLimit = ( limit + 1 ) / 2;
		boolean[] composite = new boolean[halfLimit];
		for ( int i = 1, p = 3; i < halfLimit; p += 2, i++ ) {
			if ( ! composite[i] ) {
				for ( int a = i + p; a < halfLimit; a += p ) {
					composite[a] = true;
				}
			}
		}
		
		List<Integer> result = Stream.of(2).limit(1).collect(Collectors.toList());
		for ( int i = 1, p = 3; i < halfLimit; p += 2, i++ ) {
			if ( ! composite[i] ) {
				result.add(p);
			}
		}
		return result;
	}
	
	private static List<Integer> primes;
	
	private static final int CERTAINTY_LEVEL = 15;
	
}

Output:

The first eight Smarandache-Wellin primes are:
 1:  index = 1     last prime = 2      number of digits = 1
 2:  index = 2     last prime = 3      number of digits = 2
 3:  index = 4     last prime = 7      number of digits = 4
 4:  index = 128   last prime = 719    number of digits = 355
 5:  index = 174   last prime = 1033   number of digits = 499
 6:  index = 342   last prime = 2297   number of digits = 1171
 7:  index = 435   last prime = 3037   number of digits = 1543
 8:  index = 1429  last prime = 11927  number of digits = 5719

The first 20 derived Smarandache-Wellin numbers which are prime:
 1:  index = 32    prime = 4194123321127
 2:  index = 72    prime = 547233879626521
 3:  index = 73    prime = 547233979727521
 4:  index = 134   prime = 13672766322929571043
 5:  index = 225   prime = 3916856106393739943689
 6:  index = 303   prime = 462696313560586013558131
 7:  index = 309   prime = 532727113760586013758133
 8:  index = 363   prime = 6430314317473636515467149
 9:  index = 462   prime = 8734722823685889120488197
10:  index = 490   prime = 9035923128899919621189209
11:  index = 495   prime = 9036023329699969621389211
12:  index = 522   prime = 9337023533410210710923191219
13:  index = 538   prime = 94374237357103109113243102223
14:  index = 624   prime = 117416265406198131121272110263
15:  index = 721   prime = 141459282456260193137317129313
16:  index = 738   prime = 144466284461264224139325131317
17:  index = 790   prime = 156483290479273277162351153339
18:  index = 852   prime = 164518312512286294233375158359
19:  index = 1087  prime = 208614364610327343341589284471
20:  index = 1188  prime = 229667386663354357356628334581

Julia

Translation of: Go

using Primes
using Printf

ordi(n) = n == 1 ? "st" : n == 2 ? "nd" : "th"

function SmarandacheWellin()
    pri = primes(12500)
    sw = ""
    pcount = 0
    i = 1
    println("The known Smarandache-Wellin primes are:")
    while pcount < 8
        sw *= string(pri[i])       
        if isprime(parse(BigInt, sw))
            pcount += 1
            le = length(sw)
            sws = le > 4 ? sw[1:20] * "..." * sw[le-19:le] : sw
            @printf("%d%s: index %4d  digits %4d  last prime %5d -> %s\n", pcount, ordi(pcount), i, le, pri[i], sws)
        end
        i += 1
    end
    println("\nThe first 20 Derived Smarandache-Wellin primes are:")
    freqs = zeros(Int, 10)
    pcount = 0
    i = 1
    while pcount < 20
        p = string(pri[i])
        for d in p 
            n = parse(Int, string(d)) + 1
            freqs[n] += 1
        end
        dsw = string(parse(BigInt, prod(map(string, freqs))))
        if isprime(parse(BigInt, dsw))
            pcount += 1
            @printf("%4d: index %4d  prime %s\n", pcount, i, dsw)
        end
        i += 1
    end
end

SmarandacheWellin()

Output:

Same as C and Go versions.

Nim

Library: Nim-Integers

import std/[bitops, math, strformat, strutils, sugar, tables]

import integers

const N = 12_000
let primes = @[2] & collect(for n in countup(3, N, 2):
                              if n.isPrime: n)

func compressed(str: string; size: int): string =
  ## Return a compressed value for long strings of digits.
  if str.len <= 2 * size: str
  else: &"{str[0..<size]}...{str[^size..^1]} ({str.len} digits)"

iterator swNumbers(): tuple[value: Integer; lastPrime: int] =
  ## Yield the SW-Numbers and the last prime added.
  var swString = ""
  for p in primes:
    swString.addInt p
    yield (newInteger(swString), p)

iterator derivedSwNumbers(): Integer =
  ## Yield the derived SW-Numbers.
  for (n, _) in swNumbers():
    var dSwString = ""
    let counts = toCountTable($n)
    for d in '0'..'9':
      dSwString.add $counts[d]
    yield newInteger(dSwString.strip(leading = true, trailing = false, {'0'}))

echo "Known eight Smarandache-Wellin numbers which are prime:"
var idx, count = 0
for (n, p)in swNumbers():
  inc idx
  if n.isPrime:
    inc count
    echo &"{count}:  index = {idx:<4}  last prime = {p:<5}  value = {compressed($n, 20)}"
    if count == 8: break

echo "\nFirst 20 derived S-W numbers which are prime:"
idx = 0
count = 0
for n in derivedSwNumbers():
  inc idx
  if n.isPrime:
    inc count
    echo &"{count:2}:  index = {idx:<4}  value = {n}"
    if count == 20: break

Output:

Known eight Smarandache-Wellin numbers which are prime:
1:  index = 1     last prime = 2      value = 2
2:  index = 2     last prime = 3      value = 23
3:  index = 4     last prime = 7      value = 2357
4:  index = 128   last prime = 719    value = 23571113171923293137...73677683691701709719 (355 digits)
5:  index = 174   last prime = 1033   value = 23571113171923293137...10131019102110311033 (499 digits)
6:  index = 342   last prime = 2297   value = 23571113171923293137...22732281228722932297 (1171 digits)
7:  index = 435   last prime = 3037   value = 23571113171923293137...30013011301930233037 (1543 digits)
8:  index = 1429  last prime = 11927  value = 23571113171923293137...11903119091192311927 (5719 digits)

First 20 derived S-W numbers which are prime:
 1:  index = 32    value = 4194123321127
 2:  index = 72    value = 547233879626521
 3:  index = 73    value = 547233979727521
 4:  index = 134   value = 13672766322929571043
 5:  index = 225   value = 3916856106393739943689
 6:  index = 303   value = 462696313560586013558131
 7:  index = 309   value = 532727113760586013758133
 8:  index = 363   value = 6430314317473636515467149
 9:  index = 462   value = 8734722823685889120488197
10:  index = 490   value = 9035923128899919621189209
11:  index = 495   value = 9036023329699969621389211
12:  index = 522   value = 9337023533410210710923191219
13:  index = 538   value = 94374237357103109113243102223
14:  index = 624   value = 117416265406198131121272110263
15:  index = 721   value = 141459282456260193137317129313
16:  index = 738   value = 144466284461264224139325131317
17:  index = 790   value = 156483290479273277162351153339
18:  index = 852   value = 164518312512286294233375158359
19:  index = 1087  value = 208614364610327343341589284471
20:  index = 1188  value = 229667386663354357356628334581

Perl

Library: ntheory

use v5.36; 
use ntheory <is_prime next_prime>;
use Lingua::EN::Numbers::Ordinate 'ordinate';

sub abbr ($d) { my $l = length $d; $l < 41 ? $d : substr($d,0,20) . '..' . substr($d,-20) . " ($l digits)" }

print "Smarandache-Wellen primes:\n";
my($p, $i, $nth, $n) = (0, -1, 0);
do {
    $p  = next_prime($p);
    $n .= $p;
    $i++;
    (++$nth and printf("%s: Index:%5d  Last prime:%6d  S-W: %s\n", ordinate $nth, $i, $p, abbr $n)) if is_prime $n;
} until $nth == 8;

sub derived ($n) {
    my %digits;
    $digits{$_}++ for split '', $n;
    join '', map { $digits{$_} // 0 } 0..9;
}

print "\nSmarandache-Wellen derived primes:\n";
($p, $i, $nth, $n) = (1, -1, 0, '');
do {
    $i++;
    $n .= ($p = next_prime($p));
    ++$nth and printf "%4s: Index:%5d  %s\n", ordinate $nth, $i, derived $n if is_prime derived $n;
} until $nth == 20;

Output:

Smarandache-Wellen primes:
1st: Index:    0  Last prime:     2  S-W: 2
2nd: Index:    1  Last prime:     3  S-W: 23
3rd: Index:    3  Last prime:     7  S-W: 2357
4th: Index:  127  Last prime:   719  S-W: 23571113171923293137..73677683691701709719 (355 digits)
5th: Index:  173  Last prime:  1033  S-W: 23571113171923293137..10131019102110311033 (499 digits)
6th: Index:  341  Last prime:  2297  S-W: 23571113171923293137..22732281228722932297 (1171 digits)
7th: Index:  434  Last prime:  3037  S-W: 23571113171923293137..30013011301930233037 (1543 digits)
8th: Index: 1428  Last prime: 11927  S-W: 23571113171923293137..11903119091192311927 (5719 digits)

Smarandache-Wellen derived primes:
 1st: Index:   31  4194123321127
 2nd: Index:   71  547233879626521
 3rd: Index:   72  547233979727521
 4th: Index:  133  13672766322929571043
 5th: Index:  224  3916856106393739943689
 6th: Index:  302  462696313560586013558131
 7th: Index:  308  532727113760586013758133
 8th: Index:  362  6430314317473636515467149
 9th: Index:  461  8734722823685889120488197
10th: Index:  489  9035923128899919621189209
11th: Index:  494  9036023329699969621389211
12th: Index:  521  9337023533410210710923191219
13th: Index:  537  94374237357103109113243102223
14th: Index:  623  117416265406198131121272110263
15th: Index:  720  141459282456260193137317129313
16th: Index:  737  144466284461264224139325131317
17th: Index:  789  156483290479273277162351153339
18th: Index:  851  164518312512286294233375158359
19th: Index: 1086  208614364610327343341589284471
20th: Index: 1187  229667386663354357356628334581

Phix

Rather slow...

with javascript_semantics
include mpfr.e
string sw = ""
mpz n = mpz_init(), dn = mpz_init()
sequence swp = {}, swdp = {}, dc = repeat(0,10)
integer np = 0
atom t0 = time(), t1 = time()+1
while length(swp)<iff(platform()=JS?5:8) do
    np += 1
    integer p = get_prime(np)
    string sp = sprintf("%d",p)
    sw &= sp
    mpz_set_str(n,sw,10)
    if mpz_prime(n) then
        swp = append(swp,{np,p,shorten(sw)})
    end if
    for c in sp do
        dc[c-'0'+1] += 1
    end for
    integer dc9 = dc[10]
    if odd(dc9) and remainder(dc9,5)!=0 then
        string swdps = join(dc,"",fmt:="%d")
        mpz_set_str(dn,swdps,10)
        if mpz_prime(dn) then
            swdp = append(swdp,{np,swdps})
        end if
    end if
    if platform()!=JS and time()>t1 then
        progress("processing prime %d...\r",{np})
        t1 = time()+1
    end if
end while
printf(1,"Smarandache-Wellin primes:\n")
for i,s in swp do
    printf(1," %d%s:",{i,ord(i)})
    printf(1," Index: %4d, Last prime %5d, %s\n",s)
end for
printf(1,"Smarandache-Wellin derived primes:\n")
for i,s in swdp do
    printf(1," %d%s:",{i,ord(i)})
    printf(1," Index: %4d, %s\n",s)
end for
?elapsed(time()-t0)

Output:

Smarandache-Wellin primes:
 1st: Index:    1, Last prime     2, 2
 2nd: Index:    2, Last prime     3, 23
 3rd: Index:    4, Last prime     7, 2357
 4th: Index:  128, Last prime   719, 23571113171923293137...73677683691701709719 (355 digits)
 5th: Index:  174, Last prime  1033, 23571113171923293137...10131019102110311033 (499 digits)
 6th: Index:  342, Last prime  2297, 23571113171923293137...22732281228722932297 (1,171 digits)
 7th: Index:  435, Last prime  3037, 23571113171923293137...30013011301930233037 (1,543 digits)
 8th: Index: 1429, Last prime 11927, 23571113171923293137...11903119091192311927 (5,719 digits)
Smarandache-Wellin derived primes:
 1st: Index:   32, 4194123321127
 2nd: Index:   72, 547233879626521
 3rd: Index:   73, 547233979727521
 4th: Index:  134, 13672766322929571043
 5th: Index:  225, 3916856106393739943689
 6th: Index:  303, 462696313560586013558131
 7th: Index:  309, 532727113760586013758133
 8th: Index:  363, 6430314317473636515467149
 9th: Index:  462, 8734722823685889120488197
 10th: Index:  490, 9035923128899919621189209
 11th: Index:  495, 9036023329699969621389211
 12th: Index:  522, 9337023533410210710923191219
 13th: Index:  538, 94374237357103109113243102223
 14th: Index:  624, 117416265406198131121272110263
 15th: Index:  721, 141459282456260193137317129313
 16th: Index:  738, 144466284461264224139325131317
 17th: Index:  790, 156483290479273277162351153339
 18th: Index:  852, 164518312512286294233375158359
 19th: Index: 1087, 208614364610327343341589284471
 20th: Index: 1188, 229667386663354357356628334581
"13 minutes and 13s"

Raku

The first seven Smarandache-Wellin primes are found in a few seconds on my system. The eighth adds over five minutes to the run time.

use Lingua::EN::Numbers;

my @primes = (^∞).grep: &is-prime;

my @Smarandache-Wellin = [\~] @primes;

sink @Smarandache-Wellin[1500]; # pre-reify for concurrency

sub derived ($n) { my %digits = $n.comb.Bag; (0..9).map({ %digits{$_} // 0 }).join }

sub abbr ($_) { .chars < 41 ?? $_ !! .substr(0,20) ~ '…' ~ .substr(*-20) ~ " ({.chars} digits)" }

say "Smarandache-Wellin primes:";
say ordinal-digit(++$,:u).fmt("%4s") ~ $_ for (^∞).hyper(:4batch).map({
    next unless (my $sw = @Smarandache-Wellin[$_]).is-prime;
    sprintf ": Index: %4d, Last prime: %5d, %s", $_, @primes[$_], $sw.&abbr
})[^8];

say "\nSmarandache-Wellin derived primes:";
say ordinal-digit(++$,:u).fmt("%4s") ~ $_ for (^∞).hyper(:8batch).map({
    next unless (my $sw = @Smarandache-Wellin[$_].&derived).is-prime;
    sprintf ": Index: %4d, %s", $_, $sw
})[^20];

Output:

Smarandache-Wellin primes:
 1ˢᵗ: Index:    0, Last prime:     2, 2
 2ⁿᵈ: Index:    1, Last prime:     3, 23
 3ʳᵈ: Index:    3, Last prime:     7, 2357
 4ᵗʰ: Index:  127, Last prime:   719, 23571113171923293137…73677683691701709719 (355 digits)
 5ᵗʰ: Index:  173, Last prime:  1033, 23571113171923293137…10131019102110311033 (499 digits)
 6ᵗʰ: Index:  341, Last prime:  2297, 23571113171923293137…22732281228722932297 (1171 digits)
 7ᵗʰ: Index:  434, Last prime:  3037, 23571113171923293137…30013011301930233037 (1543 digits)
 8ᵗʰ: Index: 1428, Last prime: 11927, 23571113171923293137…11903119091192311927 (5719 digits)

Smarandache-Wellin derived primes:
 1ˢᵗ: Index:   31, 4194123321127
 2ⁿᵈ: Index:   71, 547233879626521
 3ʳᵈ: Index:   72, 547233979727521
 4ᵗʰ: Index:  133, 13672766322929571043
 5ᵗʰ: Index:  224, 3916856106393739943689
 6ᵗʰ: Index:  302, 462696313560586013558131
 7ᵗʰ: Index:  308, 532727113760586013758133
 8ᵗʰ: Index:  362, 6430314317473636515467149
 9ᵗʰ: Index:  461, 8734722823685889120488197
10ᵗʰ: Index:  489, 9035923128899919621189209
11ᵗʰ: Index:  494, 9036023329699969621389211
12ᵗʰ: Index:  521, 9337023533410210710923191219
13ᵗʰ: Index:  537, 94374237357103109113243102223
14ᵗʰ: Index:  623, 117416265406198131121272110263
15ᵗʰ: Index:  720, 141459282456260193137317129313
16ᵗʰ: Index:  737, 144466284461264224139325131317
17ᵗʰ: Index:  789, 156483290479273277162351153339
18ᵗʰ: Index:  851, 164518312512286294233375158359
19ᵗʰ: Index: 1086, 208614364610327343341589284471
20ᵗʰ: Index: 1187, 229667386663354357356628334581

RPL

The latest versions of RPL can handle large 500-digit integers, making it possible to search for the fifth SW prime. Unfortunately, even with an emulator, the primality test takes too long.

Works with: HP version 49

« "" 2 
  1 4 ROLL FOR j
       SWAP OVER + SWAP NEXTPRIME
    NEXT DROP
» » '→SW' STO 

« →STR → swn
  « { 10 } 0 CON
    1 swn SIZE FOR j
       swn j DUP SUB STR→ 1 + DUP2 GET 1 + PUT 
    NEXT
    ""
    1 10 FOR j
       OVER j GET + 
    NEXT
    STR→ NIP
» 'DSW' STO 

« 0 → idx
  « { } 
    WHILE DUP SIZE 4 < REPEAT
       IF 'idx' INCR →SW DUP ISPRIME? THEN + ELSE DROP END
    END 
» » 'TASK1' STO 

« 0 → idx
  « { } 
    WHILE DUP SIZE 4 < REPEAT
       IF 'idx' INCR →SW DSW DUP ISPRIME? THEN + ELSE DROP END
    END 
» » 'TASK2' STO 

Output:

2: { 2 23 2357 2357111317192329313741434753596167717379838997101103107109113127131137139149151157163167173179181191193197199211223227229233239241251257263269271277281283293307311313317331337347349353359367373379383389397401409419421431433439443449457461463467479487491499503509521523541547557563569571577587593599601607613617619631641643647653659661673677683691701709719 }
1: { 4194123321127 547233879626521 547233979727521 13672766322929571043 }

Ruby

The first seven Smarandache-Wellin primes are found in about 12 seconds on my system. The eighth (not shown here) takes nine minutes.

require 'prime'
require 'openssl' # for it's faster 'prime?' method

Smarandache_Wellins = Struct.new(:index, :last_prime, :sw, :derived)

def derived(n)
  counter = Array.new(10){|i| [i, 0]}.to_h # counter is a Hash
  n.digits.tally(counter).values.join.to_i
end

def prime?(n) = OpenSSL::BN.new(n).prime?

ord = {1 => "1st", 2 => "2nd", 3 => "3rd"}
ord.default_proc = proc {|_h, k| k.to_s + "th"} # _ in _h means "not used"

smarandache_wellinses = Enumerator.new do |y|
  str = ""
  Prime.each.with_index(1) do |pr, i|
    str += pr.to_s
    sw = str.to_i
    y << Smarandache_Wellins.new(i, pr, sw, derived(sw))
  end
end

smarandache_wellins_primes = Enumerator.new do |y|
  smarandache_wellinses.rewind
  loop do
    s_w = smarandache_wellinses.next
    y << s_w if prime?(s_w.sw)
  end
end

sw_derived_primes = Enumerator.new do |y|
  smarandache_wellinses.rewind
  loop do
    sw  = smarandache_wellinses.next
    y << sw if prime?(sw.derived)
  end
end

n = 3
puts "The first #{n} Smarandache-Wellin primes are:"
puts smarandache_wellins_primes.first(n).map(&:sw)
puts "\nThe first #{n} Smarandache-Wellin derived primes are:"
puts sw_derived_primes.first(n).map(&:derived)

n = 7
puts "\nThe first #{n} Smarandache-Wellin primes are:"
smarandache_wellins_primes.first(n).each.with_index(1) do |swp, i|
  puts "%s: index %4d,  digits %4d,  last prime %5d " % [ord[i], swp.index, swp.sw.digits.size, swp.last_prime]
end

n = 20
puts "\nThe first #{n} Derived Smarandache-Wellin primes are:"
sw_derived_primes.first(n).each.with_index(1) do |swdp, i| 
  puts "%4s: index %4d, prime %s" % [ord[i], swdp.index, swdp.derived]
end

Output:

The first 3 Smarandache-Wellin primes are:
2
23
2357

The first 3 Smarandache-Wellin derived primes are:
4194123321127
547233879626521
547233979727521

The first 7 Smarandache-Wellin primes are:
1st: index    1,  digits    1,  last prime     2 
2nd: index    2,  digits    2,  last prime     3 
3rd: index    4,  digits    4,  last prime     7 
4th: index  128,  digits  355,  last prime   719 
5th: index  174,  digits  499,  last prime  1033 
6th: index  342,  digits 1171,  last prime  2297 
7th: index  435,  digits 1543,  last prime  3037 

The first 20 Derived Smarandache-Wellin primes are:
 1st: index   32, prime 4194123321127
 2nd: index   72, prime 547233879626521
 3rd: index   73, prime 547233979727521
 4th: index  134, prime 13672766322929571043
 5th: index  225, prime 3916856106393739943689
 6th: index  303, prime 462696313560586013558131
 7th: index  309, prime 532727113760586013758133
 8th: index  363, prime 6430314317473636515467149
 9th: index  462, prime 8734722823685889120488197
10th: index  490, prime 9035923128899919621189209
11th: index  495, prime 9036023329699969621389211
12th: index  522, prime 9337023533410210710923191219
13th: index  538, prime 94374237357103109113243102223
14th: index  624, prime 117416265406198131121272110263
15th: index  721, prime 141459282456260193137317129313
16th: index  738, prime 144466284461264224139325131317
17th: index  790, prime 156483290479273277162351153339
18th: index  852, prime 164518312512286294233375158359
19th: index 1087, prime 208614364610327343341589284471
20th: index 1188, prime 229667386663354357356628334581

Wren

Library: Wren-math

Library: Wren-fmt

Library: Wren-gmp

Need to use GMP here to find the 8th S-W prime in a reasonable time (35.5 seconds on my Core i7 machine).

import "./math" for Int
import "./gmp" for Mpz
import "./fmt"for Fmt

var primes = Int.primeSieve(12000)
var sw = ""
var count = 0
var i = 0
var n = Mpz.new()
System.print("The known Smarandache-Wellin primes are:")
while (count < 8) {
    sw = sw + primes[i].toString
    n.setStr(sw)
    if (n.probPrime(15) > 0) {
        count = count + 1
        Fmt.print("$r: index $4d  digits $4d  last prime $5d -> $20a", count, i+1, sw.count, primes[i], n)
    }
    i = i + 1
}

System.print("\nThe first 20 Derived Smarandache-Wellin primes are:")
var freqs = List.filled(10, 0)
count = 0
i = 0
while (count < 20) {
    var p = primes[i].toString
    for (d in p) {
        var n = Num.fromString(d)
        freqs[n] = freqs[n] + 1
    }
    var dsw = freqs.join("").trimStart("0")
    n.setStr(dsw)
    if (n.probPrime(15) > 0) {
        count = count + 1
        Fmt.print("$4r: index $4d  prime $i", count, i+1, n)
    }
    i = i + 1
}

Output:

The known Smarandache-Wellin primes are:
1st: index    1  digits    1  last prime     2 -> 2
2nd: index    2  digits    2  last prime     3 -> 23
3rd: index    4  digits    4  last prime     7 -> 2357
4th: index  128  digits  355  last prime   719 -> 23571113171923293137...73677683691701709719
5th: index  174  digits  499  last prime  1033 -> 23571113171923293137...10131019102110311033
6th: index  342  digits 1171  last prime  2297 -> 23571113171923293137...22732281228722932297
7th: index  435  digits 1543  last prime  3037 -> 23571113171923293137...30013011301930233037
8th: index 1429  digits 5719  last prime 11927 -> 23571113171923293137...11903119091192311927

The first 20 Derived Smarandache-Wellin primes are:
 1st: index   32  prime 4194123321127
 2nd: index   72  prime 547233879626521
 3rd: index   73  prime 547233979727521
 4th: index  134  prime 13672766322929571043
 5th: index  225  prime 3916856106393739943689
 6th: index  303  prime 462696313560586013558131
 7th: index  309  prime 532727113760586013758133
 8th: index  363  prime 6430314317473636515467149
 9th: index  462  prime 8734722823685889120488197
10th: index  490  prime 9035923128899919621189209
11th: index  495  prime 9036023329699969621389211
12th: index  522  prime 9337023533410210710923191219
13th: index  538  prime 94374237357103109113243102223
14th: index  624  prime 117416265406198131121272110263
15th: index  721  prime 141459282456260193137317129313
16th: index  738  prime 144466284461264224139325131317
17th: index  790  prime 156483290479273277162351153339
18th: index  852  prime 164518312512286294233375158359
19th: index 1087  prime 208614364610327343341589284471
20th: index 1188  prime 229667386663354357356628334581