# Fortunate numbers

Fortunate numbers
You are encouraged to solve this task according to the task description, using any language you may know.
Definition

A Fortunate number is the smallest integer m > 1 such that for a given positive integer n, primorial(n) + m is a prime number, where primorial(n) is the product of the first n prime numbers.

For example the first fortunate number is 3 because primorial(1) is 2 and 2 + 3 = 5 which is prime whereas 2 + 2 = 4 is composite.

After sorting and removal of any duplicates, compute and show on this page the first 8 Fortunate numbers or, if your language supports big integers, the first 50 Fortunate numbers.

## 11l

Translation of: Nim
```F isProbablePrime(n, k = 10)
I n < 2 | n % 2 == 0
R n == 2

V d = n - 1
V s = 0
L d % 2 == 0
d I/= 2
s++

assert(2 ^ s * d == n - 1)

Int nn
I n < 7FFF'FFFF
nn = Int(n)
E
nn = 7FFF'FFFF

L(_) 0 .< k
V a = random:(2 .< nn)
V x = pow(a, d, n)
I x == 1 | x == n - 1
L.continue
L(_) 0 .< s - 1
x = pow(x, 2, n)
I x == 1
R 0B
I x == n - 1
L.break
L.was_no_break
R 0B

R 1B

F is_prime(a)
I a == 2
R 1B
I a < 2 | a % 2 == 0
R 0B
L(i) (3 .. Int(sqrt(a))).step(2)
I a % i == 0
R 0B
R 1B

V primorial = BigInt(1)

V nn = 50
V lim = 75
V s = Set[Int]()
L(n) 1..
I is_prime(n)
primorial *= n
V m = 3
L
I isProbablePrime(primorial + m, 25)
L.break
m += 2
I --lim == 0
L.break

print(‘First ’nn‘ fortunate numbers:’)
L(m) sorted(Array(s))[0 .< nn]
V i = L.index
print(‘#3’.format(m), end' I (i + 1) % 10 == 0 {"\n"} E ‘ ’)```
Output:
```First 50 fortunate numbers:
3   5   7  13  17  19  23  37  47  59
61  67  71  79  89 101 103 107 109 127
151 157 163 167 191 197 199 223 229 233
239 271 277 283 293 307 311 313 331 353
373 379 383 397 401 409 419 421 439 443
```

## ALGOL 68

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

Uses Algol 68G's LONG LONG INT which has programmer specifiable precision.
Some arbitrary limits are used here - only the first 100 primorials are considered and it is assumed that the first 50 Fortunate numbers are all under 500.
Also shows the primorial associated with the Fortunate numbers.

The source of the ALGOL 68-primes library is on a separate Rosetta Code page - see the above link.

```BEGIN # find some Fortunate numbers m, m is smallest positive integer > 1    #
#                           where primorial(n) + m is prime for some n #
#                            as all primorials are even, m must be odd #

PR precision 2000 PR        # set the number of digits for LONG LONG INT #
PR read "primes.incl.a68" PR                   # include prime utilities #
INT max fortunate = 500;    # largeest fortunate number we will consider #
[]BOOL is prime = PRIMESIEVE 5 000;
[ 1 : max fortunate ]INT fortunate; FOR i TO max fortunate DO fortunate[ i ] := 0 OD;
INT primorial pos          := 0;
LONG LONG INT primorial    := 1;
INT prime pos              := 0;
WHILE primorial pos < 100 DO
WHILE NOT is prime[ prime pos +:= 1 ] DO SKIP OD;
primorial pos +:= 1;
primorial     *:= prime pos;
INT m          := 3;
WHILE NOT is probably prime( primorial + m ) AND m <= max fortunate DO m +:= 2 OD;
IF m <= max fortunate THEN
IF fortunate[ m ] = 0 THEN fortunate[ m ] := primorial pos FI
FI
OD;
print( ( "The first 50 Fortunate numbers:", newline ) );
INT f count := 0;
FOR f TO max fortunate WHILE f count < 50 DO
IF fortunate[ f ] /= 0 THEN
print( ( whole( f, -5 ) ) );
IF ( f count +:= 1 ) MOD 10 = 0 THEN print( ( newline ) ) FI
FI
OD;
print( ( "The primorial associated with the first 50 Fortunate numbers:", newline ) );
f count := 0;
FOR f TO max fortunate WHILE f count < 50 DO
IF fortunate[ f ] /= 0 THEN
print( ( whole( fortunate[ f ], -5 ) ) );
IF ( f count +:= 1 ) MOD 10 = 0 THEN print( ( newline ) ) FI
FI
OD

END```
Output:
```The first 50 Fortunate numbers:
3    5    7   13   17   19   23   37   47   59
61   67   71   79   89  101  103  107  109  127
151  157  163  167  191  197  199  223  229  233
239  271  277  283  293  307  311  313  331  353
373  379  383  397  401  409  419  421  439  443
The primorial associated with the first 50 Fortunate numbers:
1    2    3    4    6    7    5    9   14   16
10   11   13   21   19   24   20   15   18   28
22   35   31   36   30   23   39   27   32   26
34   55   54   53   50   57   62   45   52   65
73   59   42   63   56   75   66   67   37   51
```

## Arturo

```firstPrimes: select 1..100 => prime?
primorial: function [n][
product first.n: n firstPrimes
]

fortunates: []
i: 1

while [8 > size fortunates][
m: 3
pmi: primorial i
while -> not? prime? m + pmi
-> m: m+2
fortunates: unique fortunates ++ m
i: i + 1
]

print sort fortunates
```
Output:
`3 5 7 13 17 19 23 37`

## C

Translation of: Wren
Library: GMP
```#include <stdio.h>
#include <stdlib.h>
#include <stdbool.h>
#include <gmp.h>

int *primeSieve(int limit, int *length) {
int i, p, *primes;
int j, pc = 0;
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;
}
}
for (i = 0; i < limit; ++i) {
if (!c[i]) ++pc;
}
primes = (int *)malloc(pc * sizeof(int));
for (i = 0, j = 0; i < limit; ++i) {
if (!c[i]) primes[j++] = i;
}
free(c);
*length = pc;
return primes;
}

int compare(const void* a, const void* b) {
int arg1 = *(const int*)a;
int arg2 = *(const int*)b;
if (arg1 < arg2) return -1;
if (arg1 > arg2) return 1;
return 0;
}

int main() {
int i, j, f, pc, ac, limit = 379, fc = 0;
int *primes = primeSieve(limit, &pc);
int fortunates[80];
mpz_t primorial, temp;
mpz_init_set_ui(primorial, 1);
mpz_init(temp);
for (i = 0; i < pc; ++i) {
mpz_mul_ui(primorial, primorial, primes[i]);
for (j = 3; ; j += 2) {
if (mpz_probab_prime_p(temp, 15) > 0) {
fortunates[fc++] = j;
break;
}
}
}
qsort(fortunates, fc, sizeof(int), compare);
printf("After sorting, the first 50 distinct fortunate numbers are:\n");
for (i = 0, ac = 0; ac < 50; ++i) {
f = fortunates[i];
if (i > 0 && f == fortunates[i-1]) continue;
printf("%3d ", f);
++ac;
if (!(ac % 10)) printf("\n");
}
free(primes);
return 0;
}
```
Output:
```After sorting, the first 50 distinct fortunate numbers are:
3   5   7  13  17  19  23  37  47  59
61  67  71  79  89 101 103 107 109 127
151 157 163 167 191 197 199 223 229 233
239 271 277 283 293 307 311 313 331 353
373 379 383 397 401 409 419 421 439 443
```

## C#

Translation of: Java
```using System;
using System.Collections.Generic;
using System.Numerics;

public class FortunateNumbers
{
private const int CERTAINTY_LEVEL = 10;

public static void Main(string[] args)
{
var primes = PrimeSieve(400);
SortedSet<int> fortunates = new SortedSet<int>();
BigInteger primorial = BigInteger.One;

foreach (var prime in primes)
{
primorial *= prime;
int candidate = 3;
{
candidate += 2;
}
}

Console.WriteLine("The first 50 distinct fortunate numbers are:");
int count = 0;
foreach (var fortunate in fortunates)
{
if (count >= 50) break;
Console.Write(\$"{fortunate,4}{(count % 10 == 9 ? "\n" : "")}");
count++;
}
}

private static List<int> PrimeSieve(int aNumber)
{
var sieve = new bool[aNumber + 1];
var primes = new List<int>();

for (int i = 2; i <= aNumber; i++)
{
if (!sieve[i])
{
for (int j = i * i; j <= aNumber && j > 0; j += i)
{
sieve[j] = true;
}
}
}
return primes;
}
}

public static class BigIntegerExtensions
{
private static Random random = new Random();

public static bool IsProbablyPrime(this BigInteger source, int certainty)
{
if (source == 2 || source == 3)
return true;
if (source < 2 || source % 2 == 0)
return false;

BigInteger d = source - 1;
int s = 0;

while (d % 2 == 0)
{
d /= 2;
s += 1;
}

for (int i = 0; i < certainty; i++)
{
BigInteger a = RandomBigInteger(2, source - 2);
BigInteger x = BigInteger.ModPow(a, d, source);
if (x == 1 || x == source - 1)
continue;

for (int r = 1; r < s; r++)
{
x = BigInteger.ModPow(x, 2, source);
if (x == 1)
return false;
if (x == source - 1)
break;
}

if (x != source - 1)
return false;
}

return true;
}

private static BigInteger RandomBigInteger(BigInteger minValue, BigInteger maxValue)
{
if (minValue > maxValue)
throw new ArgumentException("minValue must be less than or equal to maxValue");

BigInteger range = maxValue - minValue + 1;
int length = range.ToByteArray().Length;
byte[] buffer = new byte[length];

BigInteger result;
do
{
random.NextBytes(buffer);
buffer[buffer.Length - 1] &= 0x7F; // Ensure non-negative
result = new BigInteger(buffer);
} while (result < minValue || result >= maxValue);

return result;
}
}
```
Output:
```The first 50 distinct fortunate numbers are:
3   5   7  13  17  19  23  37  47  59
61  67  71  79  89 101 103 107 109 127
151 157 163 167 191 197 199 223 229 233
239 271 277 283 293 307 311 313 331 353
373 379 383 397 401 409 419 421 439 443

```

## C++

```#include <algorithm>
#include <cstdint>
#include <iostream>
#include <set>
#include <vector>

std::vector<int32_t> prime_numbers(const int32_t& limit) {
const int32_t half_limit = ( limit % 2 == 0 ) ? limit / 2 : 1 + limit / 2;
std::vector<bool> composite(half_limit, false);
for ( int32_t i = 1, p = 3; i < half_limit; p += 2, ++i ) {
if ( ! composite[i] ) {
for ( int32_t a = i + p; a < half_limit; a += p ) {
composite[a] = true;
}
}
}

std::vector<int32_t> primes{2};
for ( int32_t i = 1, p = 3; i < half_limit; p += 2, ++i ) {
if ( ! composite[i] ) {
primes.push_back(p);
}
}
return primes;
}

bool contains(const std::vector<int32_t>& list, const int32_t& n) {
return std::find(list.begin(), list.end(), n) != list.end();
}

int main() {
std::vector<int32_t> primes = prime_numbers(250'000'000);
std::set<int32_t> fortunates;
int32_t primorial = 1;
int32_t index = 0;

while ( fortunates.size() < 8 ) {
primorial *= primes[index++];
int32_t candidate = 3;
while ( ! contains(primes, primorial + candidate) ) {
candidate += 2;
}
fortunates.emplace(candidate);
}

std::cout << "The first 8 distinct fortunate numbers are:" << std::endl;
for ( const int32_t& fortunate : fortunates ) {
std::cout << fortunate << " ";
}
}
```
```The first 8 distinct fortunate numbers are:
3 5 7 13 17 19 23 37
```

## Factor

Works with: Factor version 0.99 2021-06-02
```USING: grouping io kernel math math.factorials math.primes
math.ranges prettyprint sequences sets sorting ;

"First 50 distinct fortunate numbers:" print
75 [1,b] [
primorial dup next-prime 2dup - abs 1 =
[ next-prime ] when - abs
] map members natural-sort 50 head 10 group simple-table.
```
Output:
```First 50 distinct fortunate numbers:
3   5   7   13  17  19  23  37  47  59
61  67  71  79  89  101 103 107 109 127
151 157 163 167 191 197 199 223 229 233
239 271 277 283 293 307 311 313 331 353
373 379 383 397 401 409 419 421 439 443
```

## FreeBASIC

Use any primality testing example, the sets example, and Bubble Sort as includes for finding primes, removing duplicates, and sorting the output respectively. Coding these up again would bloat the code without being illustrative. Ditto for using a bigint library to get Fortunates after the 7th one, it's just not worth the bother.

```#include "isprime.bas"
#include "sets.bas"
#include "bubblesort.bas"

function prime(n as uinteger) as uinteger
if n = 1 then return 2
dim as integer c=1, p=3
while c<n
if isprime(p) then c+=1
p += 2
wend
return p
end function

function primorial( n as uinteger ) as ulongint
dim as ulongint ret = 1
for i as uinteger = 1 to n
ret *= prime(i)
next i
return ret
end function

function fortunate(n as uinteger) as uinteger
dim as uinteger m = 3
dim as ulongint pp = primorial(n)
while not isprime(m+pp)
m+=2
wend
return m
end function

redim as integer forts(-1)
dim as integer n = 0, m
while ubound(forts) < 6
n += 1
m = fortunate(n)
if not is_in(m, forts()) then
end if
wend

bubblesort(forts())
for n=0 to 6
print forts(n)
next n```

## Go

Translation of: Wren
Library: Go-rcu
```package main

import (
"fmt"
"math/big"
"rcu"
"sort"
)

func main() {
primes := rcu.Primes(379)
primorial := big.NewInt(1)
var fortunates []int
bPrime := new(big.Int)
for _, prime := range primes {
bPrime.SetUint64(uint64(prime))
primorial.Mul(primorial, bPrime)
for j := 3; ; j += 2 {
jj := big.NewInt(int64(j))
if bPrime.ProbablyPrime(5) {
fortunates = append(fortunates, j)
break
}
}
}
m := make(map[int]bool)
for _, f := range fortunates {
m[f] = true
}
fortunates = fortunates[:0]
for k := range m {
fortunates = append(fortunates, k)
}
sort.Ints(fortunates)
fmt.Println("After sorting, the first 50 distinct fortunate numbers are:")
for i, f := range fortunates[0:50] {
fmt.Printf("%3d ", f)
if (i+1)%10 == 0 {
fmt.Println()
}
}
fmt.Println()
}
```
Output:
```After sorting, the first 50 distinct fortunate numbers are:
3   5   7  13  17  19  23  37  47  59
61  67  71  79  89 101 103 107 109 127
151 157 163 167 191 197 199 223 229 233
239 271 277 283 293 307 311 313 331 353
373 379 383 397 401 409 419 421 439 443
```

```import Data.Numbers.Primes (primes)
import Math.NumberTheory.Primes.Testing (isPrime)
import Data.List (nub)

primorials :: [Integer]
primorials = 1 : scanl1 (*) primes

nextPrime :: Integer -> Integer
nextPrime n
| even n = head \$ dropWhile (not . isPrime) [n+1, n+3..]
| even n = nextPrime (n+1)

fortunateNumbers :: [Integer]
fortunateNumbers = (\p -> nextPrime (p + 2) - p) <\$> tail primorials
```
```λ> take 50 fortunateNumbers
[3,5,5,7,13,23,17,19,23,37,61,67,61,71,47,107,59,61,109,89,103,79,151,197,101,103,233,223,127,223,191,163,229,643,239,157,167,439,239,199,191,199,383,233,751,313,773,607,313,383]

-- unique fortunate numbers
λ> take 50 \$ nub \$ fortunateNumbers
[3,5,7,13,23,17,19,37,61,67,71,47,107,59,109,89,103,79,151,197,101,233,223,127,191,163,229,643,239,157,167,439,199,383,751,313,773,607,293,443,331,283,277,271,401,307,379,491,311,397]```

## J

```fortunate =: p -~ 4 p: 2 + p =. */ @: p: @ i. @ x:
echo 'Unique fortunate numbers'
echo _10 [\ 50 {. /:~ ~. fortunate"0 >: i. 75
```
Output:
```Unique fortunate numbers
3   5   7  13  17  19  23  37  47  59
61  67  71  79  89 101 103 107 109 127
151 157 163 167 191 197 199 223 229 233
239 271 277 283 293 307 311 313 331 353
373 379 383 397 401 409 419 421 439 443
```

## Java

```import java.math.BigInteger;
import java.util.BitSet;
import java.util.NavigableSet;
import java.util.TreeSet;

public final class FortunateNumbers {

public static void main(String[] aArgs) {
BitSet primes = primeSieve(400);
NavigableSet<Integer> fortunates = new TreeSet<Integer>();
BigInteger primorial = BigInteger.ONE;

for ( int prime = 2; prime >= 0; prime = primes.nextSetBit(prime + 1) ) {
primorial = primorial.multiply(BigInteger.valueOf(prime));
int candidate = 3;
while ( ! primorial.add(BigInteger.valueOf(candidate)).isProbablePrime(CERTAINTY_LEVEL) ) {
candidate += 2;
}
}

System.out.println("The first 50 distinct fortunate numbers are:");
for ( int i = 0; i < 50; i++ ) {
System.out.print(String.format("%4d%s", fortunates.pollFirst(), ( i % 10 == 9 ? "\n" : "" )));
}
}

private static BitSet primeSieve(int aNumber) {
BitSet sieve = new BitSet(aNumber + 1);
sieve.set(2, aNumber + 1);
for ( int i = 2; i <= Math.sqrt(aNumber); i = sieve.nextSetBit(i + 1) ) {
for ( int j = i * i; j <= aNumber; j = j + i ) {
sieve.clear(j);
}
}
return sieve;
}

private static final int CERTAINTY_LEVEL = 10;

}
```
Output:
```The first 50 distinct fortunate numbers are:
3   5   7  13  17  19  23  37  47  59
61  67  71  79  89 101 103 107 109 127
151 157 163 167 191 197 199 223 229 233
239 271 277 283 293 307 311 313 331 353
373 379 383 397 401 409 419 421 439 443

```

## jq

Works with: jq

Works with gojq, the Go implementation of jq

See Erdős-primes#jq for a suitable definition of `is_prime` as used here. This definition, however, is insufficiently efficient for calculating more than the first few values of fortunate(n). Here we define `fortunates(\$limit)` to be the array of length \$limit comprised of the distinct values of fortunate(n) for successive values of n.

```def primes:
2, range(3; infinite; 2) | select(is_prime);

# generate an infinite stream of primorials
def primorials:
foreach primes as \$p (1; .*\$p; .);

# Emit a sorted array of the first \$limit distinct fortunate numbers
# generated in order of the primoridials
def fortunates(\$limit):
label \$out
| foreach primorials as \$p ([];
first( range(3; infinite; 2) | select(\$p + . | is_prime)) as \$q
| . + [\$q] | unique;
if length >= \$limit then ., break \$out else empty end);

fortunates(10)```
Output:
```[3,5,7,13,17,19,23,37,61,67]
```

## Julia

```using Primes

primorials(N) = accumulate(*, primes(N), init = big"1")

primorial = primorials(800)

fortunate(n) = nextprime(primorial[n] + 2) - primorial[n]

println("After sorting, the first 50 distinct fortunate numbers are:")
foreach(p -> print(rpad(last(p), 5), first(p) % 10 == 0 ? "\n" : ""),
(map(fortunate, 1:100) |> unique |> sort!)[begin:50] |> enumerate)
```
Output:
```After sorting, the first 50 distinct fortunate numbers are:
3    5    7    13   17   19   23   37   47   59
61   67   71   79   89   101  103  107  109  127
151  157  163  167  191  197  199  223  229  233
239  271  277  283  293  307  311  313  331  353
373  379  383  397  401  409  419  421  439  443
```

## Mathematica/Wolfram Language

```ClearAll[primorials]
primorials[n_] := Times @@ Prime[Range[n]]
vals = Table[
primor = primorials[i];
s = NextPrime[primor];
t = NextPrime[s];
Min[DeleteCases[{s - primor, t - primor}, 1]]
,
{i, 100}
];
TakeSmallest[DeleteDuplicates[vals], 50]
```
Output:
`{3,5,7,13,17,19,23,37,47,59,61,67,71,79,89,101,103,107,109,127,151,157,163,167,191,197,199,223,229,233,239,271,277,283,293,307,311,313,331,353,373,379,383,397,401,409,419,421,439,443}`

## Nim

Library: bignum

Nim doesn’t provide a standard module to deal with big integers. So, we have chosen to use the third party module “bignum” which provides functions to easily find primes and check if a number is prime.

```import algorithm, sequtils, strutils
import bignum

const
N = 50      # Number of fortunate numbers.
Lim = 75    # Number of primorials to compute.

iterator primorials(lim: Positive): Int =
var prime = newInt(2)
var primorial = newInt(1)
for _ in 1..lim:
primorial *= prime
prime = prime.nextPrime()
yield primorial

var list: seq[int]
for p in primorials(Lim):
var m = 3
while true:
if probablyPrime(p + m, 25) != 0:
break
inc m, 2

list.sort()
list = list.deduplicate(true)
if list.len < N:
quit "Not enough values. Wanted \$1, got \$2.".format(N, list.len), QuitFailure
list.setLen(N)
echo "First \$# fortunate numbers:".format(N)
for i, m in list:
stdout.write (\$m).align(3), if (i + 1) mod 10 == 0: '\n' else: ' '
```
Output:
```First 50 fortunate numbers:
3   5   7  13  17  19  23  37  47  59
61  67  71  79  89 101 103 107 109 127
151 157 163 167 191 197 199 223 229 233
239 271 277 283 293 307 311 313 331 353
373 379 383 397 401 409 419 421 439 443```

## Perl

Library: ntheory
```use strict;
use warnings;
use List::Util <first uniq>;
use ntheory qw<pn_primorial is_prime>;

my \$upto = 50;
my @candidates;
for my \$p ( map { pn_primorial(\$_) } 1..2*\$upto ) {
push @candidates, first { is_prime(\$_ + \$p) } 2..100*\$upto;
}

my @fortunate = sort { \$a <=> \$b } uniq grep { is_prime \$_ } @candidates;

print "First \$upto distinct fortunate numbers:\n" .
(sprintf "@{['%6d' x \$upto]}", @fortunate) =~ s/(.{60})/\$1\n/gr;
```
Output:
```First 50 distinct fortunate numbers:
3     5     7    13    17    19    23    37    47    59
61    67    71    79    89   101   103   107   109   127
151   157   163   167   191   197   199   223   229   233
239   271   277   283   293   307   311   313   331   353
373   379   383   397   401   409   419   421   439   443```

## Phix

```with javascript_semantics
include mpfr.e
mpz primorial = mpz_init(1),
pj = mpz_init()
sequence fortunates = {}
for p=1 to 75 do
mpz_mul_si(primorial,primorial,get_prime(p))
integer j = 3
while not mpz_prime(pj) do
j = j + 2
end while
fortunates &= j
end for
fortunates = unique(deep_copy(fortunates))[1..50]
fortunates = join_by(apply(true,sprintf,{{"%3d"},fortunates}),1,10)
printf(1,"The first 50 distinct fortunate numbers are:\n%s\n",{fortunates})
```
Output:
```The first 50 distinct fortunate numbers are:
3     5     7    13    17    19    23    37    47    59
61    67    71    79    89   101   103   107   109   127
151   157   163   167   191   197   199   223   229   233
239   271   277   283   293   307   311   313   331   353
373   379   383   397   401   409   419   421   439   443
```

## Python

Library: sympy
```from sympy.ntheory.generate import primorial
from sympy.ntheory import isprime

def fortunate_number(n):
'''Return the fortunate number for positive integer n.'''
# Since primorial(n) is even for all positive integers n,
# it suffices to search for the fortunate numbers among odd integers.
i = 3
primorial_ = primorial(n)
while True:
if isprime(primorial_ + i):
return i
i += 2

fortunate_numbers = set()
for i in range(1, 76):

# Extract the first 50 numbers.
first50 = sorted(list(fortunate_numbers))[:50]

print('The first 50 fortunate numbers:')
print(('{:<3} ' * 10).format(*(first50[:10])))
print(('{:<3} ' * 10).format(*(first50[10:20])))
print(('{:<3} ' * 10).format(*(first50[20:30])))
print(('{:<3} ' * 10).format(*(first50[30:40])))
print(('{:<3} ' * 10).format(*(first50[40:])))
```
Output:
```The first 50 fortunate numbers:
3   5   7   13  17  19  23  37  47  59
61  67  71  79  89  101 103 107 109 127
151 157 163 167 191 197 199 223 229 233
239 271 277 283 293 307 311 313 331 353
373 379 383 397 401 409 419 421 439 443 ```

## Raku

Limit of 75 primorials to get first 50 unique fortunates is arbitrary, found through trial and error.

```my @primorials = [\*] grep *.is-prime, ^∞;

say display :title("First 50 distinct fortunate numbers:\n"),
(squish sort @primorials[^75].hyper.map: -> \$primorial {
(2..∞).first: (* + \$primorial).is-prime
})[^50];

sub display (\$list, :\$cols = 10, :\$fmt = '%6d', :\$title = "{+\$list} matching:\n") {
cache \$list;
\$title ~ \$list.batch(\$cols)».fmt(\$fmt).join: "\n"
}
```
Output:
```First 50 distinct fortunate numbers:
3      5      7     13     17     19     23     37     47     59
61     67     71     79     89    101    103    107    109    127
151    157    163    167    191    197    199    223    229    233
239    271    277    283    293    307    311    313    331    353
373    379    383    397    401    409    419    421    439    443```

## REXX

For this task's requirement,   finding the 8th fortunate number requires running this REXX program in a 64-bit address
space.   It is CPU intensive as there is no   isPrime   BIF for the large (possible) primes generated.

```/*REXX program finds/displays fortunate numbers  N,  where  N  is specified (default=8).*/
numeric digits 12
parse arg n cols .                               /*obtain optional argument from the CL.*/
if    n=='' |    n==","  then    n=  8           /*Not specified?  Then use the default.*/
if cols=='' | cols==","  then cols= 10           /* "      "         "   "   "     "    */
call genP n**2                                   /*build array of semaphores for primes.*/
pp.= 1
do i=1  for n+1;   im= i - 1;    pp.i= pp.im * @.i   /*calculate primorial numbers*/
end   /*i*/
i=i-1;  call genp pp.i + 1000
title= ' fortunate numbers'
w= 10                                            /*maximum width of a number in any col.*/
say ' index │'center(title, 1 + cols*(w+1)     )
say '───────┼'center(""   , 1 + cols*(w+1), '─')
found= 0;                           idx= 1       /*number of fortunate (so far) & index.*/
!!.= 0;                             maxFN= 0     /*(stemmed)  array of fortunate numbers*/
do j=1  until found==n;     pt= pp.j     /*search for fortunate numbers in range*/
pt= pp.j                                 /*get the precalculated primorial prime*/
do m=3  by 2;  t= pt + m    /*find  M  that satisfies requirement. */
if !.t==''  then leave      /*Is !.t prime?  Then we found a good M*/
end   /*m*/
if !!.m  then iterate                    /*Fortunate # already found?  Then skip*/
!!.m= 1;      found= found + 1           /*assign fortunate number;  bump count.*/
maxFN= max(maxFN, t)                     /*obtain max fortunate # for displaying*/
end   /*j*/
\$=;                                 finds= 0     /*\$:  line of output;    FINDS:  count.*/
do k=1  for maxFN;  if \!!.k  then iterate /*show the fortunate numbers we found. */
finds= finds + 1                           /*bump the  count of numbers (for \$).  */
c= commas(k)                               /*maybe add commas to the number.      */
\$= \$  right(c, max(w, length(c) ) )        /*add a nice prime ──► list, allow big#*/
if found//cols\==0  then iterate           /*have we populated a line of output?  */
say center(idx, 7)'│'  substr(\$, 2);   \$=  /*display what we have so far  (cols). */
idx= idx + cols                            /*bump the  index  count for the output*/
end   /*k*/

if \$\==''  then say center(idx, 7)"│"  substr(\$, 2)  /*possible display residual output.*/
say '───────┴'center(""   , 1 + cols*(w+1), '─')     /*display the foot separator.      */
say
say 'Found '       commas(found)      title
exit 0                                           /*stick a fork in it,  we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
commas: parse arg ?;  do jc=length(?)-3  to 1  by -3; ?=insert(',', ?, jc); end;  return ?
/*──────────────────────────────────────────────────────────────────────────────────────*/
genP:        @.1=2; @.2=3; @.3=5; @.4=7;  @.5=11 /*define some low primes.              */
!.=0;  !.2=;  !.3=;  !.5=;  !.7=;   !.11=  /*   "     "   "    "     semaphores.  */
#= 5;  sq.#= @.#**2   /*squares of low primes.*/
do j=@.#+2  by 2  to arg(1)              /*find odd primes from here on.        */
parse var j '' -1 _;     if _==5  then iterate       /*J ÷ by 5 ?               */
if j//3==0  then iterate;  if j//7==0  then iterate  /*" "  " 3?;    J ÷ by 7 ? */
do k=5  while sq.k<=j             /* [↓]  divide by the known odd primes.*/
if j // @.k == 0  then iterate j  /*Is  J ÷ X?  Then not prime.     ___  */
end   /*k*/                       /* [↑]  only process numbers  ≤  √ J   */
#= #+1;    @.#= j;    sq.#= j*j;  !.j=   /*bump # of Ps; assign next P;  P²; P# */
end          /*j*/;               return
```

output

```2nd prime generation took 580.41 seconds.
index │                                               fortunate numbers
───────┼───────────────────────────────────────────────────────────────────────────────────────────────────────────────
1   │          3          5          7         13         17         19         23          37
───────┴───────────────────────────────────────────────────────────────────────────────────────────────────────────────

Found  8  fortunate numbers
```

## Ruby

```require "gmp"

primorials = Enumerator.new do |y|
cur = prod = 1
loop {y << prod *= (cur = GMP::Z(cur).nextprime)}
end

limit = 50
fortunates = []
while fortunates.size < limit*2 do
prim = primorials.next
fortunates << (GMP::Z(prim+2).nextprime - prim)
fortunates = fortunates.uniq.sort
end

p fortunates[0, limit]
```
Output:
```[3, 5, 7, 13, 17, 19, 23, 37, 47, 59, 61, 67, 71, 79, 89, 101, 103, 107, 109, 127, 151, 157, 163, 167, 191, 197, 199, 223, 229, 233, 239, 271, 277, 283, 293, 307, 311, 313, 331, 353, 373, 379, 383, 397, 401, 409, 419, 421, 439, 443]
```

## Sidef

```func fortunate(n) {
var P = n.pn_primorial
2..Inf -> first {|m| P+m -> is_prob_prime }
}

var limit = 50
var uniq = Set()
var all = []

for (var n = 1; uniq.len < 2*limit; ++n) {
var m = fortunate(n)
all << m
uniq << m
}

say "Fortunate numbers for n = 1..#{limit}:"
say all.first(limit)

say "\n#{limit} Fortunate numbers, sorted with duplicates removed:"
say uniq.sort.first(limit)
```
Output:
```Fortunate numbers for n = 1..50:
[3, 5, 7, 13, 23, 17, 19, 23, 37, 61, 67, 61, 71, 47, 107, 59, 61, 109, 89, 103, 79, 151, 197, 101, 103, 233, 223, 127, 223, 191, 163, 229, 643, 239, 157, 167, 439, 239, 199, 191, 199, 383, 233, 751, 313, 773, 607, 313, 383, 293]

50 Fortunate numbers, sorted with duplicates removed:
[3, 5, 7, 13, 17, 19, 23, 37, 47, 59, 61, 67, 71, 79, 89, 101, 103, 107, 109, 127, 151, 157, 163, 167, 191, 197, 199, 223, 229, 233, 239, 271, 277, 283, 293, 307, 311, 313, 331, 353, 373, 379, 383, 397, 401, 409, 419, 421, 439, 443]
```

## Wren

Library: Wren-math
Library: Wren-big
Library: Wren-seq
Library: Wren-fmt
```import "./math" for Int
import "./big" for BigInt
import "./seq" for Lst
import "./fmt" for Fmt

var primes = Int.primeSieve(379)
var primorial = BigInt.one
var fortunates = []
for (prime in primes) {
primorial = primorial * prime
var j = 3
while (true) {
if ((primorial + j).isProbablePrime(5)) {
break
}
j = j + 2
}
}
fortunates = Lst.distinct(fortunates).sort()
System.print("After sorting, the first 50 distinct fortunate numbers are:")
Fmt.tprint("\$3d", fortunates[0..49], 10)
```
Output:
```After sorting, the first 50 distinct fortunate numbers are:
3   5   7  13  17  19  23  37  47  59
61  67  71  79  89 101 103 107 109 127
151 157 163 167 191 197 199 223 229 233
239 271 277 283 293 307 311 313 331 353
373 379 383 397 401 409 419 421 439 443
```