Woodall numbers

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The Woodall numbers, sometimes called Riesel numbers, and also called Cullen numbers of the second kind, are numbers of the form

Wn := n  2 n − 1, n ≥ 1,

studied by Allan J. C. Cunningham and H. J. Woodall in 1917, following James Cullen’s previous study, in 1905, of numbers of the form n  2 n  +  1.

A003261 Woodall (or Riesel) numbers

Wn

: numbers of the form

n  2 n  −  1, n   ≥   1

.

{1, 7, 23, 63, 159, 383, 895, 2047, 4607, 10239, 22527, 49151, 106495, 229375, 491519, 1048575, 2228223, 4718591, 9961471, 20971519, 44040191, 92274687, 192937983, 402653183, 838860799, 1744830463, 3623878655, 7516192767, ...}

Woodall primes

Although almost all Woodall numbers are composite, it is conjectured that the set of Woodall primes is infinite. (The same can be said for the Cullen numbers.)

A050918 Woodall primes: primes of form

k  2 k  −  1

.

{7, 23, 383, 32212254719, 2833419889721787128217599, 195845982777569926302400511, 4776913109852041418248056622882488319, 1307960347852357218937346147315859062783, ...}

A002234 Numbers

n

such that the Woodall number

n  2 n  −  1

is prime.

{2, 3, 6, 30, 75, 81, 115, 123, 249, 362, 384, 462, 512, 751, 822, 5312, 7755, 9531, 12379, 15822, 18885, 22971, 23005, 98726, 143018, 151023, 667071, 1195203, 1268979, 1467763, 2013992, 2367906, 3752948, 17016602, ...}

Generalized Woodall numbers

A?????? Generalized Woodall numbers

Wn, b

: numbers of the form

n  b n  −  1, b   ≥   2, n   ≥   b  −  1.

{1, 7, 17, 23, 63, 80, 159, 191, 323, 383, 895, 1023, 1214, 2047, 2499, 4373, 4607, 5119, 10239, 15308, 15624, 22527, 24575, 38879, 49151, 52487, 93749, 106495, 114687, 177146, 229375, 279935, 491519, 524287, 546874, 590489, 705893, 1048575, 1948616, 1959551, 2228223, ...}

A210340 Generalized Woodall primes: any primes that can be written in the form

n  b n  −  1, b   ≥   3, n   ≥   b  −  1.

(The Woodall primes for

b = 2

are given in A050918.)

{17, 191, 4373, 5119, 524287, 590489, 3124999, 14680063, 3758096383, 6973568801, 34867844009, 85449218749, 824633720831, 1099999999999, 1618481116086271, 11577835060199423, 14999999999999999, 29311444762388081, 73123168801259519, ...}

See also

• Cullen numbers

  Cn

  : numbers of the form

  n  2 n + 1, n   ≥   1

  . (A002064)
• Generalized Cullen numbers

  Cn, b

  : numbers of the form

  n  b n  +  1, b   ≥   2, n   ≥   b  −  1.

External links

• Weisstein, Eric W., Woodall Number, from MathWorld—A Wolfram Web Resource.
• Giovanni Resta, Woodall numbers—numbersaplenty.com.

• Weisstein, Eric W., Woodall Prime, from MathWorld—A Wolfram Web Resource.
• Welcome to the Cullen/Woodall Prime Search—PrimeGrid.
• Chris K. Caldwell, Woodall prime—Prime Pages.

• Paul Leyland, Generalized Cullen and Woodall numbers.
• Robert Munafo, Generalized Cullen and Woodall Numbers.
• Welcome (back) to the Generalized Cullen/Woodall Prime Search—PrimeGrid.