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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
assistant
```
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
```