Cullen and Woodall numbers

Cullen and Woodall numbers is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.

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

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

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

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

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

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