Chemistry - Woodall number

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In mathematics, a Woodall number or Riesel number is a natural number of the form n · 2ⁿ − 1 (written W_n). 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, ... . 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 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.

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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Categories: Integer sequences