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# 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.

## Factor

Works with: Factor version 0.99 2021-06-02
`USING: grouping io kernel math math.factorials math.primesmath.ranges prettyprint sequences sets sorting ; "First 50 distinct fortunate numbers:" print75 [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 pend 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 retend 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 mend function redim as integer forts(-1)dim as integer n = 0, mwhile ubound(forts) < 6    n += 1    m = fortunate(n)    if not is_in(m, forts()) then        add_to_set(m, forts())    end ifwend 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))            bPrime.Add(primorial, jj)            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 -> IntegernextPrime 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]```

## 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, strutilsimport 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:      list.add m      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), QuitFailurelist.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 primorialfrom 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):    fortunate_numbers.add(fortunate_number(i)) # 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 12parse 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.      */saysay 'Found '       commas(found)      titleexit 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 [email protected].#+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 = 50fortunates = []while fortunates.size < limit*2 do  prim = primorials.next  fortunates << (GMP::Z(prim+2).nextprime - prim)  fortunates = fortunates.uniq.sortend 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 = 50var 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-sort
Library: Wren-seq
Library: Wren-fmt
`import "/math" for Intimport "/big" for BigIntimport "/sort" for Sortimport "/seq" for Lstimport "/fmt" for Fmt var primes = Int.primeSieve(379)var primorial = BigInt.onevar fortunates = []for (prime in primes) {    primorial = primorial * prime    var j = 3    while (true) {        if ((primorial + j).isProbablePrime(5)) {            fortunates.add(j)            break        }        j = j + 2    }}fortunates = Lst.distinct(fortunates)Sort.quick(fortunates)System.print("After sorting, the first 50 distinct fortunate numbers are:")for (chunk in Lst.chunks(fortunates[0..49], 10)) Fmt.print("\$3d", chunk)`
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
```