Woodall number

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 J. 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 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

(2 | p)

is +1 and

W_{(3p − 1) / 2}

if the Jacobi symbol

(2 | p)

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.

Woodall number

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