Woodall Number

Did you know that there is a 'Woodall number'? How did this never come up in one of my math classes?

http://en.wikipedia.org/wiki/Woodall_prime


Woodall number is a natural number of the form n · 2n − 1 (written Wn).

Woodall numbers were first studied by A. J. C. Cunningham and H. J.
Woodall in 1917, inspired by James Cullen's earlier study of the
similarly-defined Cullen numbers. The first few Woodall numbers are 1,
7, 23, 63, 159, 383, 895, ... (sequence A003261 in OEIS). Woodall
numbers curiously arise in Goodstein's theorem.

Woodall numbers that are also prime numbers are called Woodall primes;
the first few exponents n for which the corresponding Woodall numbers Wn
are prime are 2, 3, 6, 30, 75, 81, 115, 123, 249, 362, 384, ...
(sequence A002234 in OEIS); the Woodall primes themselves begin with 7,
23, 383, 32212254719, ... (sequence A050918 in OEIS).

Like Cullen numbers, Woodall numbers have many divisibility properties.
For example, if p is a prime number, then p divides
W(p + 1) / 2 if the Jacobi symbol \left(\frac{2}{p}\right) is +1 and
W(3p − 1) / 2 if the Jacobi symbol \left(\frac{2}{p}\right) is −1.
It is conjectured that almost all Woodall numbers are composite; a proof
has been submitted by Suyama, but it has not been verified yet.
Nonetheless, it is also conjectured that there are infinitely many
Woodall primes.

A generalized Woodall number is defined to be a number of the form n ·
bn − 1, where n + 2 > b; if a prime can be written in this form, it is
then called a generalized Woodall prime.