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Euclid-Mullin sequence (view source)

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{{draft task}}
 
;Definition
Line 17:
[https://oeis.org/A000945 OEIS sequence A000945]
<br><br>
 
=={{header|ALGOL 68}}==
{{works with|ALGOL 68G|Any - tested with release 2.8.3.win32}}
Uses ALGOL 68G's LONG LONG INT which has programmer specifiable precission, the default is sufficient for this task.
<br>
Although the first 16 elements will all fit in 64 bits, the product exceeds 64 bits after the ninth element.
<syntaxhighlight lang="algol68">
BEGIN # find elements of the Euclid-Mullin sequence: starting from 2, #
# the next element is the smallest prime factor of 1 + the product #
# of the previous elements #
print( ( " 2" ) );
LONG LONG INT product := 2;
FROM 2 TO 16 DO
LONG LONG INT next := product + 1;
# find the first prime factor of next #
LONG LONG INT p := 3;
BOOL found := FALSE;
WHILE p * p <= next AND NOT ( found := next MOD p = 0 ) DO
p +:= 2
OD;
IF found THEN next := p FI;
print( ( " ", whole( next, 0 ) ) );
product *:= next
OD
END
</syntaxhighlight>
{{out}}
<pre>
2 3 7 43 13 53 5 6221671 38709183810571 139 2801 11 17 5471 52662739 23003
</pre>
 
=={{header|ALGOL W}}==
{{Trans|ALGOL 68|but only showing the first 8 elements as Algol W integers are limited to 32-bit.}}
<syntaxhighlight lang="algolw">
begin % find elements of the Euclid-Mullin sequence: starting from 2, %
% the next element is the smallest prime factor of 1 + the product %
% of the previous elements %
integer product;
write( "2" );
product := 2;
for i := 2 until 8 do begin
integer nextV, p;
logical found;
nextV := product + 1;
% find the first prime factor of nextV %
p := 3;
found := false;
while p * p <= nextV and not found do begin
found := nextV rem p = 0;
if not found then p := p + 2
end while_p_squared_le_nextV_and_not_found ;
if found then nextV := p;
writeon( i_w := 1, s_w := 0, " ", nextV );
product := product * nextV
end for_i
end.
</syntaxhighlight>
{{out}}
<pre>
2 3 7 43 13 53 5 6221671
</pre>
 
=={{header|AWK}}==
<syntaxhighlight lang="awk">
<lang AWK>
# syntax: GAWK -f EUCLID-MULLIN_SEQUENCE.AWK
# converted from FreeBASIC
Line 45 ⟶ 106:
exit(0)
}
</syntaxhighlight>
</lang>
{{out}}
<pre>
2 3 7 43 13 53 5 6221671
</pre>
 
Alternative version.
{{Trans|ALGOL 68}}
<syntaxhighlight lang="awk">
# find elements of the Euclid-Mullin sequence: starting from 2,
# the next element is the smallest prime factor of 1 + the product
# of the previous elements
BEGIN {
printf( "2" );
product = 2;
for( i = 2; i <= 8; i ++ )
{
nextV = product + 1;
# find the first prime factor of nextV
p = 3;
found = 0;
while( p * p <= nextV && ! ( found = nextV % p == 0 ) )
{
p += 2;
}
if( found )
{
nextV = p;
}
printf( " %d", nextV );
product *= nextV
}
}
</syntaxhighlight>
 
{{out}}
<pre>
2 3 7 43 13 53 5 6221671
</pre>
 
=={{header|BASIC}}==
==={{header|Craft Basic}}===
<syntaxhighlight lang="basic">define size = 16, em = 0
dim list[size]
 
let list[0] = 2
print 2, " ",
 
for i = 1 to 15
 
let k = 3
 
do
 
let em = 1
 
for j = 0 to i - 1
 
let em = (em * list[j]) % k
 
next j
 
let em = (em + 1) % k
 
if em = 0 then
 
let list[i] = k
print list[i], " ",
break
 
endif
 
let k = k + 2
 
wait
 
loop
 
next i</syntaxhighlight>
 
==={{header|FreeBASIC}}===
Naive and takes forever to find the largest term, but does get there in the end.
<syntaxhighlight lang="freebasic">
dim as ulongint E(0 to 15), k
dim as integer i, em
E(0) = 2 : print 2
for i=1 to 15
k=3
do
em = 1
for j as uinteger = 0 to i-1
em = (em*E(j)) mod k
next j
em = (em + 1) mod k
if em = 0 then
E(i)=k
print E(i)
exit do
end if
k = k + 2
loop
next i</syntaxhighlight>
 
=={{header|EasyLang}}==
{{trans|AWK}}
<syntaxhighlight>
limit = 8
arr[] = [ 2 ]
write 2 & " "
for i = 2 to limit
k = 3
repeat
em = 1
for j = 1 to i - 1
em = em * arr[j] mod k
.
em = (em + 1) mod k
until em = 0
k += 2
.
arr[] &= k
write k & " "
.
</syntaxhighlight>
 
=={{header|F_Sharp|F#}}==
<langsyntaxhighlight lang="fsharp">
//Euclid-Mullin sequence. Nigel Galloway: October 29th., 2021
let(|Prime|_|)(n,g)=if Open.Numeric.Primes.MillerRabin.IsProbablePrime &g then Some(n*g,n*g+1I) else None
let n=Seq.unfold(fun(n,g)->match n,g with Prime n->Some(g,n) |_->let g=Open.Numeric.Primes.Extensions.PrimeExtensions.PrimeFactors g|>Seq.item 1 in Some(g,(n*g,n*g+1I)))(1I,2I)
n|>Seq.take 16|>Seq.iter(printfn "%A")
</syntaxhighlight>
</lang>
{{out}}
<pre>
Line 78 ⟶ 259:
 
=={{header|Fermat}}==
<langsyntaxhighlight lang="fermat">Func Firstfac(n) =
j := 3;
up := Sqrt(n);
Line 94 ⟶ 275:
eu[i]:=Firstfac(1+Prod<k=1,i-1>[eu[k]]);
!(eu[i],' ');
od;</langsyntaxhighlight>
{{out}}<pre> 2 3 7 43 13 53 5 6221671 38709183810571 139 2801 11 17 5471 52662739 23003</pre>
 
=={{header|FreeBASICGo}}==
{{trans|Wren}}
Naive and takes forever to find the largest term, but does get there in the end.
{{libheader|GMP(Go wrapper)}}
<lang freebasic>
This runs in about 54 seconds which, puzzlingly, is a good bit slower than Wren even though both are using GMP and the Pollard's rho algorithm. I have no idea why.
dim as ulongint E(0 to 15), k
<syntaxhighlight lang="go">package main
dim as integer i, em
E(0) = 2 : print 2
for i=1 to 15
k=3
do
em = 1
for j as uinteger = 0 to i-1
em = (em*E(j)) mod k
next j
em = (em + 1) mod k
if em = 0 then
E(i)=k
print E(i)
exit do
end if
k = k + 2
loop
next i</lang>
 
import (
"fmt"
big "github.com/ncw/gmp"
"log"
)
 
var (
zero = big.NewInt(0)
one = big.NewInt(1)
two = big.NewInt(2)
three = big.NewInt(3)
four = big.NewInt(4)
five = big.NewInt(5)
six = big.NewInt(6)
ten = big.NewInt(10)
k100 = big.NewInt(100000)
)
 
func pollardRho(n, c *big.Int) *big.Int {
g := func(x, y *big.Int) *big.Int {
x2 := new(big.Int)
x2.Mul(x, x)
x2.Add(x2, c)
return x2.Mod(x2, y)
}
x, y, z := big.NewInt(2), big.NewInt(2), big.NewInt(1)
d := new(big.Int)
count := 0
for {
x = g(x, n)
y = g(g(y, n), n)
d.Sub(x, y)
d.Abs(d)
d.Mod(d, n)
z.Mul(z, d)
count++
if count == 100 {
d.GCD(nil, nil, z, n)
if d.Cmp(one) != 0 {
break
}
z.Set(one)
count = 0
}
}
if d.Cmp(n) == 0 {
return zero
}
return d
}
 
func smallestPrimeFactorWheel(n, max *big.Int) *big.Int {
if n.ProbablyPrime(15) {
return n
}
z := new(big.Int)
if z.Rem(n, two).Cmp(zero) == 0 {
return two
}
if z.Rem(n, three).Cmp(zero) == 0 {
return three
}
if z.Rem(n, five).Cmp(zero) == 0 {
return five
}
k := big.NewInt(7)
i := 0
inc := []*big.Int{four, two, four, two, four, six, two, six}
for z.Mul(k, k).Cmp(n) <= 0 {
if z.Rem(n, k).Cmp(zero) == 0 {
return k
}
k.Add(k, inc[i])
if k.Cmp(max) > 0 {
break
}
i = (i + 1) % 8
}
return nil
}
 
func smallestPrimeFactor(n *big.Int) *big.Int {
s := smallestPrimeFactorWheel(n, k100)
if s != nil {
return s
}
c := big.NewInt(1)
s = new(big.Int).Set(n)
for {
d := pollardRho(n, c)
if d.Cmp(zero) == 0 {
if c.Cmp(ten) == 0 {
log.Fatal("Pollard Rho doesn't appear to be working.")
}
c.Add(c, one)
} else {
// get the smallest prime factor of 'd'
factor := smallestPrimeFactorWheel(d, d)
// check whether n/d has a smaller prime factor
s = smallestPrimeFactorWheel(n.Quo(n, d), factor)
if s != nil {
if s.Cmp(factor) < 0 {
return s
} else {
return factor
}
} else {
return factor
}
}
}
}
 
func main() {
k := 19
fmt.Println("First", k, "terms of the Euclid–Mullin sequence:")
fmt.Println(2)
prod := big.NewInt(2)
z := new(big.Int)
count := 1
for count < k {
z.Add(prod, one)
t := smallestPrimeFactor(z)
fmt.Println(t)
prod.Mul(prod, t)
count++
}
}</syntaxhighlight>
 
{{out}}
<pre>
First 19 terms of the Euclid–Mullin sequence:
2
3
7
43
13
53
5
6221671
38709183810571
139
2801
11
17
5471
52662739
23003
30693651606209
37
1741
</pre>
 
=={{header|J}}==
<syntaxhighlight lang="j"> 2x (, <./@(>:&.(*/)))@[&_~ 15
2 3 7 43 13 53 5 6221671 38709183810571 139 2801 11 17 5471 52662739 23003</syntaxhighlight>
 
=={{header|Java}}==
<syntaxhighlight lang="java">
import java.math.BigInteger;
import java.util.ArrayList;
import java.util.BitSet;
import java.util.List;
import java.util.concurrent.ThreadLocalRandom;
 
public final class EuclidMullinSequence {
 
public static void main(String[] aArgs) {
primes = listPrimesUpTo(1_000_000);
System.out.println("The first 27 terms of the Euclid-Mullin sequence:");
System.out.print(2 + " ");
for ( int i = 1; i < 27; i++ ) {
System.out.print(String.format("%s%s", nextEuclidMullin(), ( i == 14 || i == 27 ) ? "\n" : " "));
}
}
private static BigInteger nextEuclidMullin() {
BigInteger smallestPrime = smallestPrimeFactor(product.add(BigInteger.ONE));
product = product.multiply(smallestPrime);
return smallestPrime;
}
private static BigInteger smallestPrimeFactor(BigInteger aNumber) {
if ( aNumber.isProbablePrime(CERTAINTY_LEVEL) ) {
return aNumber;
}
for ( BigInteger prime : primes ) {
if ( aNumber.mod(prime).signum() == 0 ) {
return prime;
}
}
BigInteger factor = pollardsRho(aNumber);
return smallestPrimeFactor(factor);
}
private static BigInteger pollardsRho(BigInteger aN) {
if ( aN.equals(BigInteger.ONE) ) {
return BigInteger.ONE;
}
if ( aN.mod(BigInteger.TWO).signum() == 0 ) {
return BigInteger.TWO;
}
final BigInteger core = new BigInteger(aN.bitLength(), random);
BigInteger x = new BigInteger(aN.bitLength(), random);
BigInteger xx = x;
BigInteger divisor = null;
do {
x = x.multiply(x).mod(aN).add(core).mod(aN);
xx = xx.multiply(xx).mod(aN).add(core).mod(aN);
xx = xx.multiply(xx).mod(aN).add(core).mod(aN);
divisor = x.subtract(xx).gcd(aN);
} while ( divisor.equals(BigInteger.ONE) );
return divisor;
}
private static List<BigInteger> listPrimesUpTo(int aLimit) {
BitSet sieve = new BitSet(aLimit + 1);
sieve.set(2, aLimit + 1);
for ( int i = 2; i * i <= aLimit; i = sieve.nextSetBit(i + 1) ) {
for ( int j = i * i; j <= aLimit; j = j + i ) {
sieve.clear(j);
}
}
List<BigInteger> result = new ArrayList<BigInteger>(sieve.cardinality());
for ( int i = 2; i >= 0; i = sieve.nextSetBit(i + 1) ) {
result.add(BigInteger.valueOf(i));
}
return result;
}
private static List<BigInteger> primes;
private static BigInteger product = BigInteger.TWO;
private static ThreadLocalRandom random = ThreadLocalRandom.current();
private static final int CERTAINTY_LEVEL = 20;
 
}
</syntaxhighlight>
{{ out }}
<pre>
The first 27 terms of the Euclid-Mullin sequence:
2 3 7 43 13 53 5 6221671 38709183810571 139 2801 11 17 5471 52662739
23003 30693651606209 37 1741 1313797957 887 71 7127 109 23 97 159227
</pre>
 
=={{header|jq}}==
'''Works with jq, gojq and jaq, that is, the C, Go and Rust implementations of jq.''' (*)
 
'''Adapted from [[#Algol_68|Algol 68]]'''
 
(*) The precision of jq and jaq is insufficient to compute more than the first nine numbers
in the sequence (i.e., beyond 38709183810571). gojq's memory consumption
becomes excessive after the first 16 numbers in the sequence are produced.
<syntaxhighlight lang=jq>
# Output: the Euclid-Mullins sequence, beginning with 2
def euclid_mullins:
foreach range(1; infinite|floor) as $i ( { product: 1 };
.next = .product + 1
# find the first prime factor of .next
| .p = 3
| .found = false
| until( .p * .p > .next or .found;
.found = ((.next % .p) == 0)
| if .found then . else .p += 2 end)
| if .found then .next = .p else . end
| .product *= .next)
| .next ;
 
# Produce 16 terms
limit(16; euclid_mullins)
</syntaxhighlight>
{{output}}
gojq supports unbounded-precision integer arithmetic
and was accordingly used to produce the following output
in accordance with the basic task requirements
Beyond this number, gojq's memory consumption becomes excessive.
 
Invocation: $ gojq -n -f euclid-mullin-sequence.algol.jq
<pre>
 
2
3
7
43
13
53
5
6221671
38709183810571
139
2801
11
17
5471
52662739
23003
</pre>
=={{header|Julia}}==
<langsyntaxhighlight lang="julia">using Primes
 
struct EuclidMullin end
Line 130 ⟶ 602:
 
println("First 16 Euclid-Mullin numbers: ", join(Iterators.take(EuclidMullin(), 16), ", "))
</langsyntaxhighlight>{{out}}
<pre>
First 16 Euclid-Mullin numbers: 2, 3, 7, 43, 13, 53, 5, 6221671, 38709183810571, 139, 2801, 11, 17, 5471, 52662739, 23003
Line 136 ⟶ 608:
 
=={{header|Mathematica}}/{{header|Wolfram Language}}==
<langsyntaxhighlight Mathematicalang="mathematica">list = {2};
Do[
prod = Times @@ list;
Line 145 ⟶ 617:
{21 - 1}
];
list</langsyntaxhighlight>
{{out}}
The first 21 numbers of the sequence:
<pre>{2, 3, 7, 43, 13, 53, 5, 6221671, 38709183810571, 139, 2801, 11, 17, 5471, 52662739, 23003, 30693651606209, 37, 1741, 1313797957, 887}</pre>
Others may be found by adjusting the range of the Do loop but it will take a while.
 
=={{header|Lua}}==
{{Trans|ALGOL W}}
<syntaxhighlight lang="lua">
-- find elements of the Euclid-Mullin sequence: starting from 2,
-- the next element is the smallest prime factor of 1 + the product
-- of the previous elements
do
io.write( "2" )
local product = 2
for i = 2, 8 do
local nextV = product + 1
-- find the first prime factor of nextV
local p = 3
local found = false
while p * p <= nextV and not found do
found = nextV % p == 0
if not found then p = p + 2 end
end
if found then nextV = p end
io.write( " ", nextV )
product = product * nextV
end
end
</syntaxhighlight>
{{out}}
<pre>
2 3 7 43 13 53 5 6221671
</pre>
 
Alternative using Pollard's Rho algorithm. Uses the iterative gcd function from the [[Greatest common divisor]] task.<br>
{{Trans|Ruby|for the Pollard's Rho algorithm}}.
Note that, as discussed on the Talk page, Pollard's Rho algorithm won't necessarily find the lowest factor, however it does for the first 16 elements.<br>
As with the other Lua sample, only 8 elements are found due to the size of some of the subsequent ones.
<syntaxhighlight lang="lua">
function gcd(a,b)
while b~=0 do
a,b=b,a%b
end
return math.abs(a)
end
function pollard_rho(n)
local x, y, d = 2, 2, 1
local g = function(x) return (x*x+1) % n end
while d == 1 do
x = g(x)
y = g(g(y))
d = gcd(math.abs(x-y),n)
end
if d == n then return d end
return math.min(d, math.floor( n/d ) )
end
 
local ar, product = {2}, 2
repeat
ar[ #ar + 1 ] = pollard_rho( product + 1 )
product = product * ar[ #ar ]
until #ar >= 8
print( table.concat(ar, " ") )
</syntaxhighlight>
{{out}}
<pre>
2 3 7 43 13 53 5 6221671
</pre>
 
=={{header|Maxima}}==
<syntaxhighlight lang="maxima">
euclid_mullin(n):=if n=1 then 2 else ifactors(1+product(euclid_mullin(i),i,1,n-1))[1][1]$
 
/* Test case */
makelist(euclid_mullin(k),k,16);
</syntaxhighlight>
{{out}}
<pre>
[2,3,7,43,13,53,5,6221671,38709183810571,139,2801,11,17,5471,52662739,23003]
</pre>
 
=={{header|MiniScript}}==
{{Trans|Lua}}
<syntaxhighlight lang="miniscript">
// find elements of the Euclid-Mullin sequence: starting from 2,
// the next element is the smallest prime factor of 1 + the product
// of the previous elements
seq = [2]
product = 2
for i in range( 2, 8 )
nextV = product + 1
// find the first prime factor of nextV
p = 3
found = false
while p * p <= nextV and not found
found = nextV % p == 0
if not found then p = p + 2
end while
if found then nextV = p
seq.push( nextV )
product = product * nextV
end for
print seq.join( " ")
</syntaxhighlight>
{{out}}
<pre>
2 3 7 43 13 53 5 6221671
</pre>
 
=={{header|Nim}}==
{{trans|Wren}}
{{libheader|integers}}
<syntaxhighlight lang="Nim">import integers
 
let
Zero = newInteger()
One = newInteger(1)
Two = newInteger(2)
Three = newInteger(3)
Five = newInteger(5)
Ten = newInteger(10)
Max = newInteger(100000)
None = newInteger(-1)
 
proc pollardRho(n, c: Integer): Integer =
 
template g(x: Integer): Integer = (x * x + c) mod n
 
var
x = newInteger(2)
y = newInteger(2)
z = newInteger(1)
d = Max + 1
count = 0
while true:
x = g(x)
y = g(g(y))
d = abs(x - y) mod n
z *= d
inc count
if count == 100:
d = gcd(z, n)
if d != One: break
z = newInteger(1)
count = 0
result = if d == n: Zero else: d
 
template isEven(n: Integer): bool = isZero(n and 1)
 
proc smallestPrimeFactorWheel(n: Integer): Integer =
if n.isPrime(5): return n
if n.isEven: return Two
if isZero(n mod 3): return Three
if isZero(n mod 5): return Five
var k = newInteger(7)
var i = 0
const Inc = [4, 2, 4, 2, 4, 6, 2, 6]
while k * k <= n:
if isZero(n mod k): return k
k += Inc[i]
if k > Max: return None
i = (i + 1) mod 8
 
proc smallestPrimeFactor(n: Integer): Integer =
var n = n
result = smallestPrimeFactorWheel(n)
if result != None: return
var c = One
result = newInteger(n)
while n > Max:
var d = pollardRho(n, c)
if d.isZero:
if c == Ten:
quit "Pollard Rho doesn't appear to be working.", QuitFailure
inc c
else:
# Can't be sure PR will find the smallest prime factor first.
result = min(result, d)
n = n div d
if n.isPrime(2):
return min(result, n)
 
proc main() =
var k = 19
echo "First ", k, " terms of the Euclid–Mullin sequence:"
echo 2
var prod = newInteger(2)
var count = 1
while count < k:
let t = smallestPrimeFactor(prod + One)
echo t
prod *= t
inc count
 
main()
</syntaxhighlight>
{{out}}
The first 16 numbers are displayed instantly, but it took about 9 seconds on my (moderately powerful) laptop to get the next three. I gave up for the 20th number.
<pre>First 19 terms of the Euclid–Mullin sequence:
2
3
7
43
13
53
5
6221671
38709183810571
139
2801
11
17
5471
52662739
23003
30693651606209
37
1741
</pre>
 
=={{header|PARI/GP}}==
<langsyntaxhighlight lang="parigp">E=vector(16)
E[1]=2
for(i=2,16,E[i]=factor(prod(n=1,i-1,E[n])+1)[1,1])
print(E)</langsyntaxhighlight>
{{out}}<pre>[2, 3, 7, 43, 13, 53, 5, 6221671, 38709183810571, 139, 2801, 11, 17, 5471, 52662739, 23003]</pre>
 
=={{header|Perl}}==
{{libheader|ntheory}}
<langsyntaxhighlight lang="perl">use strict;
use warnings;
use feature 'say';
Line 169 ⟶ 856:
 
say "First sixteen: @Euclid_Mullin[ 0..15]";
say "Next eleven: @Euclid_Mullin[16..26]";</langsyntaxhighlight>
{{out}}
<pre>First sixteen: 2 3 7 43 13 53 5 6221671 38709183810571 139 2801 11 17 5471 52662739 23003
Line 175 ⟶ 862:
 
=={{header|Phix}}==
<!--<langsyntaxhighlight Phixlang="phix">(phixonline)-->
<span style="color: #008080;">with</span> <span style="color: #008080;">javascript_semantics</span>
<span style="color: #7060A8;">requires</span><span style="color: #0000FF;">(</span><span style="color: #008000;">"1.0.1"</span><span style="color: #0000FF;">)</span> <span style="color: #000080;font-style:italic;">-- (added mpz_set_v())</span>
Line 189 ⟶ 876:
<span style="color: #008080;">end</span> <span style="color: #008080;">while</span>
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"The first 16 Euclid-Mulin numbers: %s\n"</span><span style="color: #0000FF;">,{</span><span style="color: #7060A8;">join</span><span style="color: #0000FF;">(</span><span style="color: #000000;">res</span><span style="color: #0000FF;">)})</span>
<!--</langsyntaxhighlight>-->
{{out}}
<pre>
Line 198 ⟶ 885:
 
=={{header|Python}}==
<langsyntaxhighlight lang="python">""" Rosetta code task: Euclid-Mullin_sequence """
 
from primePy import primes
Line 212 ⟶ 899:
GEN = euclid_mullin()
print('First 16 Euclid-Mullin numbers:', ', '.join(str(next(GEN)) for _ in range(16)))
</langsyntaxhighlight>{{out}}
<pre>
First 16 Euclid-Mullin numbers: 2, 3, 7, 43, 13, 53, 5, 6221671, 38709183810571, 139, 2801, 11, 17, 5471, 52662739, 23003
Line 219 ⟶ 906:
=={{header|Raku}}==
 
<syntaxhighlight lang="raku" perl6line>use Prime::Factor;
 
my @Euclid-Mullin = 2, { state $i = 1; (1 + [×] @Euclid-Mullin[^$i++]).&prime-factors.min } … *;
 
put 'First sixteen: ', @Euclid-Mullin[^16];</langsyntaxhighlight>
{{out}}
<pre>First sixteen: 2 3 7 43 13 53 5 6221671 38709183810571 139 2801 11 17 5471 52662739 23003</pre>
 
=={{header|Ring}}==
{{Trans|Lua}}
<syntaxhighlight lang="ring">
// find elements of the Euclid-Mullin sequence: starting from 2,
// the next element is the smallest prime factor of 1 + the product
// of the previous elements
see "2"
product = 2
for i = 2 to 8
nextV = product + 1
// find the first prime factor of nextV
p = 3
found = false
while p * p <= nextV and not found
found = ( nextV % p ) = 0
if not found p = p + 2 ok
end
if found nextV = p ok
see " " + nextV
product = product * nextV
next
</syntaxhighlight>
{{out}}
<pre>
2 3 7 43 13 53 5 6221671
</pre>
 
=={{header|RPL}}==
{{works with|Halcyon Calc|4.2.7}}
{| class="wikitable"
! RPL code
! Comment
|-
|
≪
'''IF''' # 1d DUP2 AND ≠ '''THEN''' DROP # 2d
'''ELSE IF''' DUP 3 DUP2 / * == '''THEN''' DROP # 3d
'''ELSE''' DUP B→R √ → divm
≪ 4 5 divm '''FOR''' n
'''IF''' OVER n DUP2 / * ==
'''THEN''' SWAP DROP n R→B SWAP divm 'n' STO '''END'''
6 SWAP - DUP '''STEP''' DROP
≫ '''END END'''
≫ ‘'''bDIV1'''’ STO
≪
DUP SIZE 1 1 ROT '''FOR''' j OVER j GET * '''NEXT'''
1 + '''bDIV1''' +
≫ ''''NXTEM'''' STO
|
'''bDIV1''' ''( #m -- #first_divisor )''
is #2 a divisor ?
is #3 a divisor ?
otherwise get sqrt(m)
d = 4 ; for n = 5 to sqrt(m)
if n divides m
replace m by n and prepare loop exit
d = 6 - d ; n += d
'''NXTEM''' ''( { #EM(1) .. #EM(n) } -- { #EM(1) .. #EM(n+1) } )''
get EM(1)*..*EM(n)
get least prime factor of 1+EM(1)*..*EM(n) and add to list
|}
{{in}}
<pre>
≪ { # 2 } WHILE DUP SIZE ≤ 16 REPEAT NXTEM END ≫ EVAL
</pre>
The emulator's watchdog timer prevents checking the primality of EM(9) = # 38709183810571d. Even if this device stayed idle, EM(10) could not be calculated, since the product of all earlier elements is more than 64 bits long.
{{out}}
<pre>
1: { # 2d # 3d # 7d # 43d # 13d # 53d # 5d # 6221671d }
</pre>
=={{header|Ruby}}==
<syntaxhighlight lang="ruby">def pollard_rho(n)
x, y, d = 2, 2, 1
g = proc{|x|(x*x+1) % n}
while d == 1 do
x = g[x]
y = g[g[y]]
d = (x-y).abs.gcd(n)
end
return d if d == n
[d, n/d].compact.min
end
 
ar = [2]
ar << pollard_rho(ar.inject(&:*)+1) until ar.size >= 16
puts ar.join(", ")
</syntaxhighlight>
{{out}}
<pre>2, 3, 7, 43, 13, 53, 5, 6221671, 38709183810571, 139, 2801, 11, 17, 5471, 52662739, 23003
</pre>
=={{header|SETL}}==
<syntaxhighlight lang="setl">program euclid_mullin;
print(2);
product := 2;
loop for i in [2..16] do
next := smallest_factor(product + 1);
product *:= next;
print(next);
end loop;
 
op smallest_factor(n);
if even n then return 2; end if;
d := 3;
loop while d*d <= n do
if n mod d=0 then return d; end if;
d +:= 2;
end loop;
return n;
end op;
end program;</syntaxhighlight>
{{out}}
<pre>2
3
7
43
13
53
5
6221671
38709183810571
139
2801
11
17
5471
52662739
23003</pre>
 
=={{header|Sidef}}==
<langsyntaxhighlight lang="ruby">func f(n) is cached {
return 2 if (n == 1)
lpf(1 + prod(1..^n, {|k| f(k) }))
Line 234 ⟶ 1,054:
 
say f.map(1..16)
say f.map(17..27)</langsyntaxhighlight>
{{out}}
<pre>
Line 244 ⟶ 1,064:
===Wren-cli===
This uses the [https://en.wikipedia.org/wiki/Pollard%27s_rho_algorithm Pollard Rho algorithm] to try and speed up the factorization of the 15th element but overall time still slow at around 32 seconds.
<langsyntaxhighlight ecmascriptlang="wren">import "./big" for BigInt
 
var zero = BigInt.zero
Line 250 ⟶ 1,070:
var two = BigInt.two
var ten = BigInt.ten
var max k100 = BigInt.new(100000)
 
var pollardRho = Fn.new { |n, c|
var g = Fn.new { |x, y| (x*x + c) % n }
var x = two
var y = two
var z = one
var d = max + one
var count = 0
while (true) {
x = g.call(x, n)
y = g.call(g.call(y, n), n)
d = (x - y).abs % n
z = z * d
count = count + 1
if (count == 100) {
d = BigInt.gcd(z, n)
if (d != one) break
z = one
count = 0
}
}
if (d == n) return zero
return d
}
 
var smallestPrimeFactorWheel = Fn.new { |n, max|
if (n.isProbablePrime(5)) return n
if (n % 2 == zero) return BigInt.two
Line 293 ⟶ 1,089:
 
var smallestPrimeFactor = Fn.new { |n|
var s = smallestPrimeFactorWheel.call(n, k100)
if (s) return s
var c = one
swhile =(true) n{
var d = BigInt.pollardRho(n, 2, c)
while (n > max) {
var d = pollardRho.call(n, c)
if (d == 0) {
if (c == ten) Fiber.abort("Pollard Rho doesn't appear to be working.")
c = c + one
} else {
// can't be sure PR will findget the smallest prime factor firstof 'd'
svar factor = BigIntsmallestPrimeFactorWheel.mincall(sd, d)
n// =check nwhether n/ d has a smaller prime factor
ifs (n.isProbablePrime(2))= return BigIntsmallestPrimeFactorWheel.mincall(sn/d, nfactor)
return s ? BigInt.min(s, factor) : factor
}
}
return s
}
 
Line 322 ⟶ 1,117:
prod = prod * t
count = count + 1
}</syntaxhighlight>
} </lang>
 
{{out}}
Line 345 ⟶ 1,140:
</pre>
<br>
 
===Embedded===
{{libheader|Wren-gmp}}
This finds the first 16 in 0.11 seconds andbut the next 311 intakes around6 39minutes 17 seconds. I gave up after that as it would take too long for the Pollard's Rho algorithm to find any more.
 
<lang ecmascript>/* euclid_mullin_gmp.wren */
If we could assume that Pollard's Rho will always find the smallest prime factor (or a multiple thereof) first, then this would bring the runtime down to 44 seconds and still produce the correct answers for this particular task. However, in general it is not safe to assume that - see discussion in Talk Page.
<syntaxhighlight lang="wren">/* Euclid_mullin_sequence_2.wren */
 
import "./gmp" for Mpz
 
var maxk100 = Mpz.from(100000)
 
var smallestPrimeFactorWheelsmallestPrimeFactorTrial = Fn.new { |n, max|
if (n.probPrime(15) > 0) return n
ifvar (n.isEven)k return= Mpz.twoone
if (n.isDivisibleUi(3)) return Mpz.three
if (n.isDivisibleUi(5)) return Mpz.five
var k = Mpz.from(7)
var i = 0
var inc = [4, 2, 4, 2, 4, 6, 2, 6]
while (k * k <= n) {
if (nk.isDivisible(k)) return knextPrime
k.add(inc[i])
if (k > max) return null
i =if (i + 1n.isDivisible(k)) %return 8k
}
}
 
var smallestPrimeFactor = Fn.new { |n|
var s = smallestPrimeFactorWheelsmallestPrimeFactorTrial.call(n, k100)
if (s) return s
var c = Mpz.one
swhile = n.copy(true) {
while (n > max) {
var d = Mpz.pollardRho(n, 2, c)
if (d.isZero) {
Line 381 ⟶ 1,172:
c.inc
} else {
// can't be sure PR will findget the smallest prime factor firstof 'd'
svar factor = smallestPrimeFactorTrial.mincall(d, d)
// check whether n.div(/d) has a smaller prime factor
ifs (n.probPrime(5)= > 0) return MpzsmallestPrimeFactorTrial.mincall(sn/d, nfactor)
return s ? Mpz.min(s, factor) : factor
}
}
return s
}
 
var k = 1927
System.print("First %(k) terms of the Euclid–Mullin sequence:")
System.print(2)
Line 400 ⟶ 1,191:
prod.mul(t)
count = count + 1
}</langsyntaxhighlight>
 
{{out}}
As Wren-cli plus threeeleven more:
<pre>
30693651606209
37
1741
1313797957
887
71
7127
109
23
97
159227
</pre>
PureFox
9,476

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