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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
-
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
: numbers of the form
.
-
{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
.
-
{7, 23, 383, 32212254719, 2833419889721787128217599, 195845982777569926302400511, 4776913109852041418248056622882488319, 1307960347852357218937346147315859062783, ...}
A002234 Numbers
such that the Woodall number
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 : 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
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
External links