Cullen and Woodall numbers

Cullen and Woodall numbers
You are encouraged to solve this task according to the task description, using any language you may know.

A Cullen number is a number of the form n × 2ⁿ + 1 where n is a natural number.

A Woodall number is very similar. It is a number of the form n × 2ⁿ - 1 where n is a natural number.

So for each n the associated Cullen number and Woodall number differ by 2.

Woodall numbers are sometimes referred to as Riesel numbers or Cullen numbers of the second kind.

Cullen primes are Cullen numbers that are prime. Similarly, Woodall primes are Woodall numbers that are prime.

It is common to list the Cullen and Woodall primes by the value of n rather than the full evaluated expression. They tend to get very large very quickly. For example, the third Cullen prime, n == 4713, has 1423 digits when evaluated.

Task

• Write procedures to find Cullen numbers and Woodall numbers.

• Use those procedures to find and show here, on this page the first 20 of each.

Stretch

• Find and show the first 5 Cullen primes in terms of n.

• Find and show the first 12 Woodall primes in terms of n.

See also

• OEIS:A002064 - Cullen numbers: a(n) = n*2^n + 1

• OEIS:A003261 - Woodall (or Riesel) numbers: n*2^n - 1

• OEIS:A005849 - Indices of prime Cullen numbers: numbers k such that k*2^k + 1 is prime

• OEIS:A002234 - Numbers k such that the Woodall number k*2^k - 1 is prime

ALGOL 68

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

Uses Algol 68Gs LONG LONG INT for long integers. The number of digits must be specified and appears to affect the run time as larger sies are specified. This sample only shows the first two Cullen primes as the time taken to find the third is rather long.

Library: ALGOL 68-primes

BEGIN # find Cullen and Woodall numbers and determine which are prime #
      # a Cullen number n is n2^2 + 1, Woodall number is n2^n - 1     #
    PR read "primes.incl.a68" PR                  # include prime utilities #
    PR precision 800 PR # set number of digits for Algol 68G LONG LONG INT #
    # returns the nth Cullen number #
    OP CULLEN = ( INT n )LONG LONG INT: n * LONG LONG INT(2)^n + 1;
    # returns the nth Woodall number #
    OP WOODALL = ( INT n )LONG LONG INT: CULLEN n - 2;

    # show the first 20 Cullen numbers #
    print( ( "1st 20 Cullen numbers:" ) );
    FOR n TO 20 DO
        print( ( " ", whole( CULLEN n, 0 ) ) )
    OD;
    print( ( newline ) );
    # show the first 20 Woodall numbers #
    print( ( "1st 20 Woodall numbers:" ) );
    FOR n TO 20 DO
        print( ( " ", whole( WOODALL n, 0 ) ) )
    OD;
    print( ( newline ) );
    BEGIN # first 2 Cullen primes #
        print( ( "Index of the 1st 2 Cullen primes:" ) );
        LONG LONG INT power of 2 := 1;
        INT prime count := 0;
        FOR n WHILE prime count < 2 DO
            power of 2 *:= 2;
            LONG LONG INT c n = ( n * power of 2 ) + 1;
            IF is probably prime( c n ) THEN
                prime count +:= 1;
                print( ( " ", whole( n, 0 ) ) )
            FI
        OD;
        print( ( newline ) )
    END;
    BEGIN # first 12 Woodall primes #
        print( ( "Index of the 1st 12 Woodall primes:" ) );
        LONG LONG INT power of 2 := 1;
        INT prime count := 0;
        FOR n WHILE prime count < 12 DO
            power of 2 *:= 2;
            LONG LONG INT w n = ( n * power of 2 ) - 1;
            IF is probably prime( w n ) THEN
                prime count +:= 1;
                print( ( " ", whole( n, 0 ) ) )
            FI
        OD;
        print( ( newline ) )
    END
END

Output:

1st 20 Cullen numbers: 3 9 25 65 161 385 897 2049 4609 10241 22529 49153 106497 229377 491521 1048577 2228225 4718593 9961473 20971521
1st 20 Woodall numbers: 1 7 23 63 159 383 895 2047 4607 10239 22527 49151 106495 229375 491519 1048575 2228223 4718591 9961471 20971519
Index of the 1st 2 Cullen primes: 1 141
Index of the 1st 12 Woodall primes: 2 3 6 30 75 81 115 123 249 362 384 462

Arturo

cullen: function [n]->
    inc n * 2^n

woodall: function [n]->
    dec n * 2^n

print ["First 20 cullen numbers:" join.with:" " to [:string] map 1..20 => cullen]
print ["First 20 woodall numbers:" join.with:" " to [:string] map 1..20 => woodall]

Output:

First 20 cullen numbers: 3 9 25 65 161 385 897 2049 4609 10241 22529 49153 106497 229377 491521 1048577 2228225 4718593 9961473 20971521 
First 20 woodall numbers: 1 7 23 63 159 383 895 2047 4607 10239 22527 49151 106495 229375 491519 1048575 2228223 4718591 9961471 20971519

AWK

# syntax: GAWK -f CULLEN_AND_WOODALL_NUMBERS.AWK
BEGIN {
    start = 1
    stop = 20
    printf("Cullen %d-%d:",start,stop)
    for (n=start; n<=stop; n++) {
      printf(" %d",n*(2^n)+1)
    }
    printf("\n")
    printf("Woodall %d-%d:",start,stop)
    for (n=start; n<=stop; n++) {
      printf(" %d",n*(2^n)-1)
    }
    printf("\n")
    exit(0)
}

Output:

Cullen 1-20: 3 9 25 65 161 385 897 2049 4609 10241 22529 49153 106497 229377 491521 1048577 2228225 4718593 9961473 20971521
Woodall 1-20: 1 7 23 63 159 383 895 2047 4607 10239 22527 49151 106495 229375 491519 1048575 2228223 4718591 9961471 20971519

BASIC

BASIC256

Translation of: FreeBASIC

print "First 20 Cullen numbers:"

for n = 1 to 20
	num = n * (2^n)+1
	print int(num); " ";
next

print : print
print "First 20 Woodall numbers:"

for n = 1 to 20
	num = n * (2^n)-1
	print int(num); " ";
next n
end

Output:

Igual que la entrada de FreeBASIC.

FreeBASIC

Dim As Uinteger n, num
Print "First 20 Cullen numbers:"

For n = 1 To 20
    num = n * (2^n)+1
    Print num; " ";
Next

Print !"\n\nFirst 20 Woodall numbers:"

For n = 1 To 20
    num = n * (2^n)-1
    Print num; " ";
Next n
Sleep

Output:

First 20 Cullen numbers:
3 9 25 65 161 385 897 2049 4609 10241 22529 49153 106497 229377 491521 1048577 2228225 4718593 9961473 20971521 

First 20 Woodall numbers:
1 7 23 63 159 383 895 2047 4607 10239 22527 49151 106495 229375 491519 1048575 2228223 4718591 9961471 20971519

PureBasic

OpenConsole()
PrintN("First 20 Cullen numbers:")

For n.i = 1 To 20
  num = n * Pow(2, n)+1
  Print(Str(num) + " ")
Next

PrintN(#CRLF$ + "First 20 Woodall numbers:")

For n.i = 1 To 20
  num = n * Pow(2, n)-1
  Print(Str(num) + " ")
Next n

PrintN(#CRLF$ + "--- terminado, pulsa RETURN---"): Input()
CloseConsole()

Output:

Igual que la entrada de FreeBASIC.

QBasic

Works with: QBasic version 1.1

Works with: QuickBasic version 4.5

Works with: True BASIC

Translation of: FreeBASIC

DIM num AS LONG ''comment this line for True BASIC 
PRINT "First 20 Cullen numbers:"

FOR n = 1 TO 20
    LET num = n * (2 ^ n) + 1
    PRINT num;
NEXT n

PRINT
PRINT
PRINT "First 20 Woodall numbers:"

FOR n = 1 TO 20
    LET num = n * (2 ^ n) - 1
    PRINT num;
NEXT n
END

Output:

Igual que la entrada de FreeBASIC.

True BASIC

Works with: QBasic

Translation of: FreeBASIC

REM DIM num AS LONG               !uncomment this line for QBasic
PRINT "First 20 Cullen numbers:"

FOR n = 1 TO 20
    LET num = n * (2 ^ n) + 1
    PRINT num;
NEXT n

PRINT 
PRINT
PRINT "First 20 Woodall numbers:"

FOR n = 1 TO 20
    LET num = n * (2 ^ n) - 1
    PRINT num;
NEXT n
END

Output:

Igual que la entrada de FreeBASIC.

Yabasic

print "First 20 Cullen numbers:"

for n = 1 to 20
    num = n * (2^n)+1
    print num, " ";
next

print "\n\nFirst 20 Woodall numbers:"

for n = 1 to 20
    num = n * (2^n)-1
    print num, " ";
next n
print
end

Output:

Igual que la entrada de FreeBASIC.

F#

// Cullen and Woodall numbers. Nigel Galloway: January 14th., 2022
let Cullen,Woodall=let fG n (g:int)=(bigint g)*2I**g+n in fG 1I, fG -1I
Seq.initInfinite((+)1>>Cullen)|>Seq.take 20|>Seq.iter(printf "%A "); printfn ""
Seq.initInfinite((+)1>>Woodall)|>Seq.take 20|>Seq.iter(printf "%A "); printfn ""
Seq.initInfinite((+)1)|>Seq.filter(fun n->let mutable  n=Woodall n in Open.Numeric.Primes.MillerRabin.IsProbablePrime &n)|>Seq.take 12|>Seq.iter(printf "%A "); printfn ""
Seq.initInfinite((+)1)|>Seq.filter(fun n->let mutable  n=Cullen n in Open.Numeric.Primes.MillerRabin.IsProbablePrime &n)|>Seq.take 5|>Seq.iter(printf "%A "); printfn ""

Output:

3 9 25 65 161 385 897 2049 4609 10241 22529 49153 106497 229377 491521 1048577 2228225 4718593 9961473 20971521 
1 7 23 63 159 383 895 2047 4607 10239 22527 49151 106495 229375 491519 1048575 2228223 4718591 9961471 20971519 
2 3 6 30 75 81 115 123 249 362 384 462 
1 141 4713 5795 6611

Factor

Works with: Factor version 0.99 2022-04-03

USING: arrays kernel math math.vectors prettyprint ranges
sequences ;

20 [1..b] [ dup 2^ * 1 + ] map dup 2 v-n 2array simple-table.

Output:

3 9 25 65 161 385 897 2049 4609 10241 22529 49153 106497 229377 491521 1048577 2228225 4718593 9961473 20971521
1 7 23 63 159 383 895 2047 4607 10239 22527 49151 106495 229375 491519 1048575 2228223 4718591 9961471 20971519

Go

Library: GMP(Go wrapper)

package main

import (
    "fmt"
    big "github.com/ncw/gmp"
)

func cullen(n uint) *big.Int {
    one := big.NewInt(1)
    bn := big.NewInt(int64(n))
    res := new(big.Int).Lsh(one, n)
    res.Mul(res, bn)
    return res.Add(res, one)
}

func woodall(n uint) *big.Int {
    res := cullen(n)
    return res.Sub(res, big.NewInt(2))
}

func main() {
    fmt.Println("First 20 Cullen numbers (n * 2^n + 1):")
    for n := uint(1); n <= 20; n++ {
        fmt.Printf("%d ", cullen(n))
    }

    fmt.Println("\n\nFirst 20 Woodall numbers (n * 2^n - 1):")
    for n := uint(1); n <= 20; n++ {
        fmt.Printf("%d ", woodall(n))
    }

    fmt.Println("\n\nFirst 5 Cullen primes (in terms of n):")
    count := 0
    for n := uint(1); count < 5; n++ {
        cn := cullen(n)
        if cn.ProbablyPrime(15) {
            fmt.Printf("%d ", n)
            count++
        }
    }

    fmt.Println("\n\nFirst 12 Woodall primes (in terms of n):")
    count = 0
    for n := uint(1); count < 12; n++ {
        cn := woodall(n)
        if cn.ProbablyPrime(15) {
            fmt.Printf("%d ", n)
            count++
        }
    }
    fmt.Println()
}

Output:

First 20 Cullen numbers (n * 2^n + 1):
3 9 25 65 161 385 897 2049 4609 10241 22529 49153 106497 229377 491521 1048577 2228225 4718593 9961473 20971521 

First 20 Woodall numbers (n * 2^n - 1):
1 7 23 63 159 383 895 2047 4607 10239 22527 49151 106495 229375 491519 1048575 2228223 4718591 9961471 20971519 

First 5 Cullen primes (in terms of n):
1 141 4713 5795 6611 

First 12 Woodall primes (in terms of n):
2 3 6 30 75 81 115 123 249 362 384 462 

Haskell

findCullen :: Int -> Integer
findCullen n = toInteger ( n * 2 ^ n + 1 )

cullens :: [Integer]
cullens = map findCullen [1 .. 20]

woodalls :: [Integer]
woodalls = map (\i -> i - 2 ) cullens

main :: IO ( )
main = do
   putStrLn "First 20 Cullen numbers:"
   print cullens
   putStrLn "First 20 Woodall numbers:"
   print woodalls

Output:

First 20 Cullen numbers:
[3,9,25,65,161,385,897,2049,4609,10241,22529,49153,106497,229377,491521,1048577,2228225,4718593,9961473,20971521]
First 20 Woodall numbers:
[1,7,23,63,159,383,895,2047,4607,10239,22527,49151,106495,229375,491519,1048575,2228223,4718591,9961471,20971519]

J

cullen=:  {{ y* 1+2x^y }}
woodall=: {{ y*_1+2x^y }}

Task example:

   cullen 1+i.20
3 10 27 68 165 390 903 2056 4617 10250 22539 49164 106509 229390 491535 1048592 2228241 4718610 9961491 20971540
   woodall 1+i.20
1 6 21 60 155 378 889 2040 4599 10230 22517 49140 106483 229362 491505 1048560 2228207 4718574 9961453 20971500

Julia

Translation of: Raku

using Lazy
using Primes

cullen(n, two = BigInt(2)) = n * two^n + 1
woodall(n, two = BigInt(2)) = n * two^n - 1
primecullens = @>> Lazy.range() filter(n -> isprime(cullen(n)))
primewoodalls = @>> Lazy.range() filter(n -> isprime(woodall(n)))

println("First 20 Cullen numbers: ( n × 2**n + 1)\n", [cullen(n, 2) for n in 1:20]) # A002064
println("First 20 Woodall numbers: ( n × 2**n - 1)\n", [woodall(n, 2) for n in 1:20]) # A003261
println("\nFirst 5 Cullen primes: (in terms of n)\n",  take(5, primecullens))  # A005849
println("\nFirst 12 Woodall primes: (in terms of n)\n", Int.(collect(take(12, primewoodalls)))) # A002234

Output:

First 20 Cullen numbers: ( n × 2**n + 1)
[3, 9, 25, 65, 161, 385, 897, 2049, 4609, 10241, 22529, 49153, 106497, 229377, 491521, 1048577, 2228225, 4718593, 9961473, 20971521]
First 20 Woodall numbers: ( n × 2**n - 1)
[1, 7, 23, 63, 159, 383, 895, 2047, 4607, 10239, 22527, 49151, 106495, 229375, 491519, 1048575, 2228223, 4718591, 9961471, 20971519]

First 5 Cullen primes: (in terms of n)
List: (1 141 4713 5795 6611)

First 12 Woodall primes: (in terms of n)
[2, 3, 6, 30, 75, 81, 115, 123, 249, 362, 384, 462]

Lua

function T(t) return setmetatable(t, {__index=table}) end
table.range = function(t,n) local s=T{} for i=1,n do s[i]=i end return s end
table.map = function(t,f) local s=T{} for i=1,#t do s[i]=f(t[i]) end return s end

function cullen(n) return (n<<n)+1 end
print("First 20 Cullen numbers:")
print(T{}:range(20):map(cullen):concat(" "))

function woodall(n) return (n<<n)-1 end
print("First 20 Woodall numbers:")
print(T{}:range(20):map(woodall):concat(" "))

Output:

First 20 Cullen numbers:
3 9 25 65 161 385 897 2049 4609 10241 22529 49153 106497 229377 491521 1048577 2228225 4718593 9961473 20971521
First 20 Woodall numbers:
1 7 23 63 159 383 895 2047 4607 10239 22527 49151 106495 229375 491519 1048575 2228223 4718591 9961471 20971519

Mathematica/Wolfram Language

ClearAll[CullenNumber, WoodallNumber]
SetAttributes[{CullenNumber, WoodallNumber}, Listable]
CullenNumber[n_Integer] := n 2^n + 1
WoodallNumber[n_Integer] := n 2^n - 1

CullenNumber[Range[20]]
WoodallNumber[Range[20]]

cps = {};
Do[
 If[PrimeQ[CullenNumber[i]],
  AppendTo[cps, i];
  If[Length[cps] >= 5, Break[]]
  ]
 ,
 {i, 1, \[Infinity]}
 ]
cps

wps = {};
Do[
  If[PrimeQ[WoodallNumber[i]],
   AppendTo[wps, i];
   If[Length[wps] >= 12, Break[]]
   ]
  ,
  {i, 1, \[Infinity]}
  ];
wps

Output:

{3, 9, 25, 65, 161, 385, 897, 2049, 4609, 10241, 22529, 49153, 106497, 229377, 491521, 1048577, 2228225, 4718593, 9961473, 20971521}
{1, 7, 23, 63, 159, 383, 895, 2047, 4607, 10239, 22527, 49151, 106495, 229375, 491519, 1048575, 2228223, 4718591, 9961471, 20971519}
{1, 141, 4713, 5795, 6611}
{2, 3, 6, 30, 75, 81, 115, 123, 249, 362, 384, 462}

Perl

Library: ntheory

use strict;
use warnings;
use bigint;
use ntheory 'is_prime';
use constant Inf  => 1e10;

sub cullen {
    my($n,$c) = @_;
    ($n * 2**$n) + $c;
}

my($m,$n);

($m,$n) = (20,0);
print "First $m Cullen numbers:\n";
print do { $n < $m ? (++$n and cullen($_,1) . ' ') : last } for 1 .. Inf;

($m,$n) = (20,0);
print "\n\nFirst $m Woodall numbers:\n";
print do { $n < $m ? (++$n and cullen($_,-1) . ' ') : last } for 1 .. Inf;

($m,$n) = (5,0);
print "\n\nFirst $m Cullen primes: (in terms of n)\n";
print do { $n < $m ? (!!is_prime(cullen $_,1) and ++$n and "$_ ") : last } for 1 .. Inf;

($m,$n) = (12,0);
print "\n\nFirst $m Woodall primes: (in terms of n)\n";
print do { $n < $m ? (!!is_prime(cullen $_,-1) and ++$n and "$_ ") : last } for 1 .. Inf;

Output:

First 20 Cullen numbers:
3 9 25 65 161 385 897 2049 4609 10241 22529 49153 106497 229377 491521 1048577 2228225 4718593 9961473 20971521

First 20 Woodall numbers:
1 7 23 63 159 383 895 2047 4607 10239 22527 49151 106495 229375 491519 1048575 2228223 4718591 9961471 20971519

First 5 Cullen primes: (in terms of n)
1 141 4713 5795 6611

First 12 Woodall primes: (in terms of n)
2 3 6 30 75 81 115 123 249 362 384 462

Phix

with javascript_semantics
atom t0 = time()
include mpfr.e
 
procedure cullen(mpz r, integer n)
    mpz_ui_pow_ui(r,2,n)
    mpz_mul_si(r,r,n)
    mpz_add_si(r,r,1)
end procedure
 
procedure woodall(mpz r, integer n)
    cullen(r,n)
    mpz_sub_si(r,r,2)
end procedure

sequence c = {}, w = {}
mpz z = mpz_init()
for i=1 to 20 do
    cullen(z,i)
    c = append(c,mpz_get_str(z))
    mpz_sub_si(z,z,2)
    w = append(w,mpz_get_str(z))
end for
printf(1," Cullen[1..20]:%s\nWoodall[1..20]:%s\n",{join(c),join(w)})
 
atom t1 = time()+1
c = {}
integer n = 1
while length(c)<iff(platform()=JS?2:5) do
    cullen(z,n)
    if mpz_prime(z) then c = append(c,sprint(n)) end if
    n += 1
    if time()>t1 and platform()!=JS then
        progress("c(%d) [needs to get to 6611], %d found\r",{n,length(c)})
        t1 = time()+2
    end if
end while
if platform()!=JS then progress("") end if
printf(1,"First 5 Cullen primes (in terms of n):%s\n",{join(c)})
w = {}
n = 1
while length(w)<12 do
    woodall(z,n)
    if mpz_prime(z) then w = append(w,sprint(n)) end if
    n += 1
end while
printf(1,"First 12 Woodall primes (in terms of n):%s\n",{join(w)})
?elapsed(time()-t0)

Output:

 Cullen[1..20]:3 9 25 65 161 385 897 2049 4609 10241 22529 49153 106497 229377 491521 1048577 2228225 4718593 9961473 20971521
Woodall[1..20]:1 7 23 63 159 383 895 2047 4607 10239 22527 49151 106495 229375 491519 1048575 2228223 4718591 9961471 20971519
First 5 Cullen primes (in terms of n):1 141 4713 5795 6611
First 12 Woodall primes (in terms of n):2 3 6 30 75 81 115 123 249 362 384 462
"34.4s"

Note the time given is for desktop/Phix 64bit, for comparison the Julia entry took about 20s on the same box. On 32-bit it is nearly 5 times slower (2 minutes and 38s) and hence under pwa/p2js in a browser (which is inherently 32bit) it is limited to the first 2 cullen primes only, but manages that in 0.4s.

Python

print("working...")
print("First 20 Cullen numbers:")

for n in range(1,20):
    num = n*pow(2,n)+1
    print(str(num),end= " ")

print()
print("First 20 Woodall numbers:")

for n in range(1,20):
    num = n*pow(2,n)-1
    print(str(num),end=" ")

print()
print("done...")

Output:

working...
First 20 Cullen numbers:
3 9 25 65 161 385 897 2049 4609 10241 22529 49153 106497 229377 491521 1048577 2228225 4718593 9961473 
First 20 Woodall numbers:
1 7 23 63 159 383 895 2047 4607 10239 22527 49151 106495 229375 491519 1048575 2228223 4718591 9961471 
done...

Bit Shift

Translation of: Quackery

def cullen(n): return((n<<n)+1)
	
def woodall(n): return((n<<n)-1)

print("First 20 Cullen numbers:")
for i in range(1,20):
	print(cullen(i),end=" ")
print()
print()
print("First 20 Woodall numbers:")
for i in range(1,20): 
	print(woodall(i),end=" ")
print()

Output:

Same as Quackery.

Quackery

  [ dup << 1+ ]  is cullen  ( n --> n )

  [ dup << 1 - ] is woodall ( n --> n )

  say "First 20 Cullen numbers:" cr
  20 times [ i^ 1+ cullen echo sp ] cr
  cr
  say "First 20 Woodall numbers:" cr
  20 times [ i^ 1+ woodall echo sp ] cr

Output:

First 20 Cullen numbers:
3 9 25 65 161 385 897 2049 4609 10241 22529 49153 106497 229377 491521 1048577 2228225 4718593 9961473 20971521 

First 20 Woodall numbers:
1 7 23 63 159 383 895 2047 4607 10239 22527 49151 106495 229375 491519 1048575 2228223 4718591 9961471 20971519 

Raku

my @cullen  = ^∞ .map: { $_ × 1 +< $_ + 1 };
my @woodall = ^∞ .map: { $_ × 1 +< $_ - 1 };

put "First 20 Cullen numbers: ( n × 2**n + 1)\n",     @cullen[1..20]; # A002064
put "\nFirst 20 Woodall numbers: ( n × 2**n - 1)\n", @woodall[1..20]; # A003261
put "\nFirst 5 Cullen primes: (in terms of n)\n",     @cullen.grep( &is-prime, :k )[^5];  # A005849
put "\nFirst 12 Woodall primes:  (in terms of n)\n", @woodall.grep( &is-prime, :k )[^12]; # A002234

Output:

First 20 Cullen numbers: ( n × 2**n + 1)
3 9 25 65 161 385 897 2049 4609 10241 22529 49153 106497 229377 491521 1048577 2228225 4718593 9961473 20971521

First 20 Woodall numbers: ( n × 2**n - 1)
1 7 23 63 159 383 895 2047 4607 10239 22527 49151 106495 229375 491519 1048575 2228223 4718591 9961471 20971519

First 5 Cullen primes: (in terms of n)
1 141 4713 5795 6611

First 12 Woodall primes:  (in terms of n)
2 3 6 30 75 81 115 123 249 362 384 462

Ring

load "stdlib.ring"

see "working..." + nl
see "First 20 Cullen numbers:" + nl

for n = 1 to 20
    num = n*pow(2,n)+1
    see "" + num + " "
next

see nl + nl + "First 20 Woodall numbers:" + nl

for n = 1 to 20
    num = n*pow(2,n)-1
    see "" + num + " "
next

see nl + "done..." + nl

Output:

working...
First 20 Cullen numbers:
3 9 25 65 161 385 897 2049 4609 10241 22529 49153 106497 229377 491521 1048577 2228225 4718593 9961473 20971521 

First 20 Woodall numbers:
1 7 23 63 159 383 895 2047 4607 10239 22527 49151 106495 229375 491519 1048575 2228223 4718591 9961471 20971519 
done...

Rust

// [dependencies]
// rug = "1.15.0"

use rug::integer::IsPrime;
use rug::Integer;

fn cullen_number(n: u32) -> Integer {
    let num = Integer::from(n);
    (num << n) + 1
}

fn woodall_number(n: u32) -> Integer {
    let num = Integer::from(n);
    (num << n) - 1
}

fn main() {
    println!("First 20 Cullen numbers:");
    let cullen: Vec<String> = (1..21).map(|x| cullen_number(x).to_string()).collect();
    println!("{}", cullen.join(" "));

    println!("\nFirst 20 Woodall numbers:");
    let woodall: Vec<String> = (1..21).map(|x| woodall_number(x).to_string()).collect();
    println!("{}", woodall.join(" "));

    println!("\nFirst 5 Cullen primes in terms of n:");
    let cullen_primes: Vec<String> = (1..)
        .filter_map(|x| match cullen_number(x).is_probably_prime(25) {
            IsPrime::No => None,
            _ => Some(x.to_string()),
        })
        .take(5)
        .collect();
    println!("{}", cullen_primes.join(" "));

    println!("\nFirst 12 Woodall primes in terms of n:");
    let woodall_primes: Vec<String> = (1..)
        .filter_map(|x| match woodall_number(x).is_probably_prime(25) {
            IsPrime::No => None,
            _ => Some(x.to_string()),
        })
        .take(12)
        .collect();
    println!("{}", woodall_primes.join(" "));
}

Output:

First 20 Cullen numbers:
3 9 25 65 161 385 897 2049 4609 10241 22529 49153 106497 229377 491521 1048577 2228225 4718593 9961473 20971521

First 20 Woodall numbers:
1 7 23 63 159 383 895 2047 4607 10239 22527 49151 106495 229375 491519 1048575 2228223 4718591 9961471 20971519

First 5 Cullen primes in terms of n:
1 141 4713 5795 6611

First 12 Woodall primes in terms of n:
2 3 6 30 75 81 115 123 249 362 384 462

Sidef

func cullen(n)  { n * (1 << n) + 1 }
func woodall(n) { n * (1 << n) - 1 }

say "First 20 Cullen numbers:"
say cullen.map(1..20).join(' ')

say "\nFirst 20 Woodall numbers:"
say woodall.map(1..20).join(' ')

say "\nFirst 5 Cullen primes: (in terms of n)"
say 5.by { cullen(_).is_prime }.join(' ')

say "\nFirst 12 Woodall primes: (in terms of n)"
say 12.by { woodall(_).is_prime }.join(' ')

Output:

First 20 Cullen numbers:
3 9 25 65 161 385 897 2049 4609 10241 22529 49153 106497 229377 491521 1048577 2228225 4718593 9961473 20971521

First 20 Woodall numbers:
1 7 23 63 159 383 895 2047 4607 10239 22527 49151 106495 229375 491519 1048575 2228223 4718591 9961471 20971519

First 5 Cullen primes: (in terms of n)
1 141 4713 5795 6611

First 12 Woodall primes: (in terms of n)
2 3 6 30 75 81 115 123 249 362 384 462

Verilog

module main;
  integer n, num;
  
  initial begin
      $display("First 20 Cullen numbers:");
      for(n = 1; n <= 20; n=n+1) 
      begin
        num = n * (2 ** n) + 1;
        $write(num, "  ");
      end
      $display("");
      $display("First 20 Woodall numbers:");
      for(n = 1; n <= 20; n=n+1) 
      begin
        num = n * (2 ** n) - 1;
        $write(num, "  ");
      end
      $finish ;
    end
endmodule

Wren

CLI

Library: Wren-big

Cullen primes limited to first 2 as very slow after that.

import "./big" for BigInt

var cullen = Fn.new { |n| (BigInt.one << n) * n + 1 }

var woodall = Fn.new { |n| cullen.call(n) - 2 }

System.print("First 20 Cullen numbers (n * 2^n + 1):")
for (n in 1..20) System.write("%(cullen.call(n)) ")

System.print("\n\nFirst 20 Woodall numbers (n * 2^n - 1):")
for (n in 1..20) System.write("%(woodall.call(n)) ")

System.print("\n\nFirst 2 Cullen primes (in terms of n):")
var count = 0
var n = 1
while (count < 2) {
    var cn = cullen.call(n)
    if (cn.isProbablePrime(5)){
        System.write("%(n) ")
        count = count + 1
    }
    n = n + 1
}

System.print("\n\nFirst 12 Woodall primes (in terms of n):")
count = 0
n = 1
while (count < 12) {
    var wn = woodall.call(n)
    if (wn.isProbablePrime(5)){
        System.write("%(n) ")
        count = count + 1
    }
    n = n + 1
}
System.print()

Output:

First 20 Cullen numbers (n * 2^n + 1):
3 9 25 65 161 385 897 2049 4609 10241 22529 49153 106497 229377 491521 1048577 2228225 4718593 9961473 20971521 

First 20 Woodall numbers (n * 2^n - 1):
1 7 23 63 159 383 895 2047 4607 10239 22527 49151 106495 229375 491519 1048575 2228223 4718591 9961471 20971519 

First 2 Cullen primes (in terms of n):
1 141 

First 12 Woodall primes (in terms of n):
2 3 6 30 75 81 115 123 249 362 384 462 

Embedded

Library: Wren-gmp

Cullen primes still slow to emerge, just over 10 seconds overall.

/* cullen_and_woodall_numbers2.wren */

import "./gmp" for Mpz

var cullen = Fn.new { |n| (Mpz.one << n) * n + 1 }

var woodall = Fn.new { |n| cullen.call(n) - 2 }

System.print("First 20 Cullen numbers (n * 2^n + 1):")
for (n in 1..20) System.write("%(cullen.call(n)) ")

System.print("\n\nFirst 20 Woodall numbers (n * 2^n - 1):")
for (n in 1..20) System.write("%(woodall.call(n)) ")

System.print("\n\nFirst 5 Cullen primes (in terms of n):")
var count = 0
var n = 1
while (count < 5) {
    var cn = cullen.call(n)
    if (cn.probPrime(15) > 0){
        System.write("%(n) ")
        count = count + 1
    }
    n = n + 1
}

System.print("\n\nFirst 12 Woodall primes (in terms of n):")
count = 0
n = 1
while (count < 12) {
    var wn = woodall.call(n)
    if (wn.probPrime(15) > 0){
        System.write("%(n) ")
        count = count + 1
    }
    n = n + 1
}
System.print()

Output:

First 20 Cullen numbers (n * 2^n + 1):
3 9 25 65 161 385 897 2049 4609 10241 22529 49153 106497 229377 491521 1048577 2228225 4718593 9961473 20971521 

First 20 Woodall numbers (n * 2^n - 1):
1 7 23 63 159 383 895 2047 4607 10239 22527 49151 106495 229375 491519 1048575 2228223 4718591 9961471 20971519 

First 5 Cullen primes (in terms of n):
1 141 4713 5795 6611 

First 12 Woodall primes (in terms of n):
2 3 6 30 75 81 115 123 249 362 384 462