Cullen Number

A number of the form

$$C_n =2^n n+1.$$

The first few are 3, 9, 25, 65, 161, 385, ... (Sloane's A002064). The only Cullen numbers $C_n$ for $n<300,000$ which are Prime are for $n=1$, 141, 4713, 5795, 6611, 18496, 32292, 32469, 59656, 90825, 262419, ... (Sloane's A005849; Ballinger). Cullen numbers are Divisible by $p=2n-1$ if $p$ is a Prime of the form $8k\pm 3$.

See also Cunningham Number, Fermat Number, Sierpinski Number of the First Kind, Woodall Number

References

Ballinger, R. ``Cullen Primes: Definition and Status.'' http://vamri.xray.ufl.edu/proths/cullen.html.

Guy, R. K. ``Cullen Numbers.'' §B20 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, p. 77, 1994.

Keller, W. ``New Cullen Primes.'' Math. Comput. 64, 1733-1741, 1995.

Leyland, P. ftp://sable.ox.ac.uk/pub/math/factors/cullen.

Ribenboim, P. The New Book of Prime Number Records. New York: Springer-Verlag, pp. 360-361, 1996.

Sloane, N. J. A. Sequences A002064/M2795 and A005849/M5401 in ``An On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995.