Woodall number

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In mathematics, a Woodall number is a natural number of the form n · 2ⁿ − 1 (written W_n). Woodall numbers were first studied by Allan 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 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 W_n are prime are 2, 3, 6, 30, 75, 81, 115, 123, 249, 362, 384, ... (sequence in OEIS); the Woodall primes themselves begin with 7, 23, 383, 32212254719, ... (sequence 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\right)] is +1 and

W_{(3p − 1) / 2} if the Jacobi symbol [\left(\frac\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.

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