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In number theory, a Woodall number (Wn) is any natural number of the form

\( W_n = n \cdot 2^n - 1 \)

for some natural number n. The first few Woodall numbers are:

1, 7, 23, 63, 159, 383, 895, … (sequence A003261 in OEIS).

History

Woodall numbers were first studied by Allan J. C. Cunningham and H. J. Woodall in 1917,[1] inspired by James Cullen's earlier study of the similarly-defined Cullen numbers. Woodall numbers curiously arise in Goodstein's theorem.

Woodall primes

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).

In 1976 Christopher Hooley showed that almost all Cullen numbers are composite.[citation needed] Hooley's proof was reworked by Hiromi Suyama to show that it works for any sequence of numbers n · 2n+a + b where a and b are integers, and in particular also for Woodall numbers. Nonetheless, it is conjectured that there are infinitely many Woodall primes.[citation needed] As of December 2007, the largest known Woodall prime is 3752948 × 23752948 − 1.[2] It has 1,129,757 digits and was found by Matthew J. Thompson in 2007 in the distributed computing project PrimeGrid.
Divisibility properties

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.

Generalized Woodall number

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.

Least n such that n*bⁿ - 1 is prime are

3, 2, 1, 1, 8, 1, 2, 1, 10, 2, 2, 1, 2, 1, 2, 167, 2, 1, 12, 1, 2, 2, 29028, 1, 2, 3, 10, 2, 26850, 1, 8, 1, 42, 2, 6, 2, 24, 1, 2, 3, 2, 1, 2, 1, 2, 2, 140, 1, 2, 2, ... (sequence A240235 in OEIS)

Dual Woodall number

Let n<0, the numerator of the number n*2ⁿ - 1 is 2ⁿ + n, the first few of them are

1, 3, 6, 11, 20, 37, 70, 135, 264, 521, 1034, 2059, 4108, 8205, 16398, 32783, 65552, 131089, 262162, 524307, 1048596, 2097173, 4194326, 8388631, 16777240, ... (sequence A006127 in OEIS)

Numbers n such that 2n + n is prime are

1, 3, 5, 9, 15, 39, 75, 81, 89, 317, 701, 735, ... (sequence A052007 in OEIS)

The dual Woodall prime themselves are

3, 11, 37, 521, 32783, 549755813927, ... (sequence A129962 in OEIS)