Woodall prime

Soon after the Reverend Cullen examined the numbers of the form n.2n+1, the numbers n.2n-1 were looked at by Cunningham and Woodall (1917). So now these numbers are called Cullen numbers: Cn=n.2n+1, and the Woodall numbers: Wn=n.2n-1. The Woodall numbers are sometimes called the Cullen numbers (of the second kind).

Woodall numbers that are prime are called Woodall primes (or Cullen primes of the second kind) It is conjectured that there are infinitely many such primes. The Woodall numbers Wn are primes for n=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 and composite for all other exponents n less than 260,000.

Like the Cunningham numbers, the Woodall numbers have many divisibility properties. For example, if p is a prime, then p divides W(p+1)/2 when the Jacobi symbol (2|p) is 1 and W(3p-1)/2 when the Jacobi symbol (2|p) is -1. Suyama appears to have shown that almost all Woodall numbers are composite [Keller95].

Generalized Woodall primes, should we wish to make such a definition, would be primes of the form n.bn-1 with n+2 > b. The reason for the restriction on the exponent n is simple, without some restriction every prime p would be a generalized Woodall because:

p = 1.(p+1)1-1.

References:

CW17
A. J. C. Cunningham and H. J. Woodall, "Factorisation of Q=(2q ± q) and q*2q ± 1," Math. Mag., 47 (1917) 1--38. [A classic paper in the history of the study of Cullen numbers. See also [Keller95]]
Guy94 (section B2)
R. K. Guy, Unsolved problems in number theory, Springer-Verlag, 1994.  New York, NY, ISBN 0-387-94289-0. MR 96e:11002 [An excellent resource! Guy briefly describes many open questions, then provides numerous references. See his newer editions of this text.]
Karst73
E. Karst, Prime factors of Cullen numbers n· 2n± 1.  In "Number Theory Tables," A. Brousseau editor, Fibonacci Assoc., 1973.  San Jose, CA, pp. 153--163,
Keller83
W. Keller, "Factors of Fermat numbers and large primes of the form k· 2n +1," Math. Comp., 41 (1983) 661-673.  MR 85b:11117
Keller95
W. Keller, "New Cullen primes," Math. Comp., 64 (1995) 1733-1741.  Supplement S39-S46.  MR 95m:11015
Ribenboim95 (p. 360-361)
P. Ribenboim, The new book of prime number records, 3rd edition, Springer-Verlag, New York, NY, 1995.  pp. xxiv+541, ISBN 0-387-94457-5. MR 96k:11112 [An excellent resource for those with some college mathematics. Basically a Guinness Book of World Records for primes with much of the relevant mathematics. The extensive bibliography is seventy-five pages.]
Riesel69
H. Riesel, "Some factors of the numbers Gn = 62n + 1 and Hn = 102n + 1," Math. Comp., 23:106 (1969) 413--415.  MR 39:6813
Robinson58
R. M. Robinson, "A report on primes of the form k· 2n + 1 and on factors of Fermat numbers," Proc. Amer. Math. Soc., 9 (1958) 673--681.  MR 20:3097