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.
- 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
11l
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)
s.add(m)
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
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
#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) {
mpz_add_ui(temp, primorial, j);
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#
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;
while (!BigInteger.Add(primorial, candidate).IsProbablyPrime(CERTAINTY_LEVEL))
{
candidate += 2;
}
fortunates.Add(candidate);
}
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])
{
primes.Add(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
EasyLang
fastfunc isprim num .
i = 2
while i <= sqrt num
if num mod i = 0
return 0
.
i += 1
.
return 1
.
fastfunc nextprim prim .
repeat
prim += 1
until isprim prim = 1
.
return prim
.
proc insert e . d[] .
for i = 1 to len d[]
if d[i] >= e
if d[i] = e
return
.
break 1
.
.
d[] &= 0
for i = len d[] downto i + 1
d[i] = d[i - 1]
.
d[i] = e
.
proc fortunates . .
maxint = pow 2 53
primorial = 1
prim = 2
repeat
primorial *= prim
until primorial + 1000 >= maxint
prim = nextprim prim
j = 3
while isprim (primorial + j) = 0
j = j + 2
.
insert j fortuns[]
.
for i to 8
write fortuns[i] & " "
.
.
fortunates
- Output:
3 5 7 13 17 19 23 37
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.
- 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
add_to_set(m, forts())
end if
wend
bubblesort(forts())
for n=0 to 6
print forts(n)
next n
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()
}
- 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
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;
}
fortunates.add(candidate);
}
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 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
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:
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: ' '
- 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
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 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
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:])))
- 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
Version 1
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
Version 2
Libraries: How to use
Library: Functions
Library: Numbers
Library: Constants
Library: Settings
Library: Abend
Library: Sequences
Version 1 does not apply an Prime() function, but uses a sieve to generate and store all primes up to a given threshold. Then for REXX you get into trouble very soon, because the primorials grow so fast in magnitude. As seen above, generating the first 8 fortunate numbers already takes 10 minutes. Also this version is quite complex and not very readable.
I have an efficient primality test, Prime() Miler-Rabin in Numbers and a procedure to generate the first n primorials, Primorials() in Sequences. From try-outs (and other entries for this task) it seems using the first 75 primorials is sufficient. The precision (numeric digits) is set accordingly. It's clear that the primality tests are key in this version. All these considerations result in below program.
include Settings
say version; say 'Fortunate numbers'; say
numeric digits 155
call GetPrimorials 75
call GenerateFortunate 75
call ShowFortunate 50
exit
GetPrimorials:
call Time('r')
arg x
say 'Get the first' x 'primorials...'
call Primorials(-x)
say Format(Time('e'),,3) 'seconds'; say
return
GenerateFortunate:
call Time('r')
arg x
say 'Generate fortunate numbers...'
work. = 0; m = 0; n = 0
do i = 1 to x
p = prmo.primorial.i
do j = 3 by 2
m = m+1
if Prime(p+j) then
leave j
end
if work.j = 0 then do
work.j = 1; n = n+1
end
end
say m 'primality tests performed'
say n 'fortunate numbers found'
say Format(Time('e'),,3) 'seconds'; say
return
ShowFortunate:
call Time('r')
arg x
say 'First' x 'fortunate numbers:'
n = 0
do i = 1
if work.i then do
n = n+1
call CharOut ,Right(i,5)
if n//10 = 0 then
say
if n = x then
leave i
end
end
say Format(Time('e'),,3) 'seconds'; say
return
include Numbers
include Sequences
include Functions
include Constants
include Abend
- Output:
REXX-ooRexx_5.0.0(MT)_64-bit 6.05 23 Dec 2022
Fortunate numbers
Get the first 75 primorials...
0.018 seconds
Generate fortunate numbers...
10227 primality tests performed
59 fortunate numbers found
233.660 seconds
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
0.001 seconds
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
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)) {
fortunates.add(j)
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