Fortunate numbers: Difference between revisions
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=={{header|Ruby}}== |
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<lang ruby>require "gmp" |
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primorials = Enumerator.new do |y| |
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cur = prod = 1 |
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loop {y << prod *= (cur = GMP::Z(cur).nextprime)} |
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end |
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limit = 50 |
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fortunates = [] |
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while fortunates.size < limit*2 do |
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prim = primorials.next |
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fortunates << (GMP::Z(prim+2).nextprime - prim) |
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fortunates = fortunates.uniq.sort |
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end |
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p fortunates[0, limit] |
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</lang> |
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{{out}} |
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<pre>[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] |
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</pre> |
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=={{header|Sidef}}== |
=={{header|Sidef}}== |
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<lang ruby>func fortunate(n) { |
<lang ruby>func fortunate(n) { |
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Revision as of 15:32, 22 November 2021
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.
- Task
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.
- Related task
- See also
- oeis:A005235 Fortunate numbers
- oeis:A046066 Fortunate numbers, sorted with duplicates removed
Factor
<lang factor>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.</lang>
- 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.
<lang freebasic>
- 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
add_to_set(m, forts())
end if
wend
bubblesort(forts()) for n=0 to 6
print forts(n)
next n</lang>
Go
<lang go>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()
}</lang>
- 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
Haskell
<lang haskell>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</lang>
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
<lang 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)
</lang>
- 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
<lang Mathematica>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]</lang>
- 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
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. <lang Nim>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:
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), 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: ' '</lang>
- 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
<lang perl>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;</lang>
- 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 mpz_add_si(pj,primorial,3) while not mpz_prime(pj) do mpz_add_si(pj,pj,2) 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
<lang python>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):
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:])))</lang>
- 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.
<lang perl6>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"
}</lang>
- 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.
<lang rexx>/*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</lang>
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
<lang 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] </lang>
- 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
<lang ruby>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)</lang>
- 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
<lang ecmascript>import "/math" for Int import "/big" for BigInt import "/sort" for Sort 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)) {
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)</lang>
- 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