Cullen and Woodall numbers

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

Julia

Translation of: Raku

<lang julia>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)))

primewoodalls(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

</lang>

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]

Raku

<lang perl6>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</lang>

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

Wren

CLI

Library: Wren-big

Cullen primes limited to first 2 as very slow after that. <lang ecmascript>import "./big" for BigInt

var cullen = Fn.new { |n|

   n = BigInt.new(n)
   return (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()</lang>

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. <lang ecmascript>/* cullen_and_woodall_numbers2.wren */

import "./gmp" for Mpz