A Cullen number is a number of the form

The first few are 3, 9, 25, 65, 161, 385, ... (OEIS A002064).

Cullen numbers are divisible by  if  is a prime of the form .

The first few prime Cullen numbers are numbers 3, 393050634124102232869567034555427371542904833, ... (OEIS A050920), corresponding to , 141, 4713, 5795, 6611, 18496, 32292, 32469, 59656, 90825, 262419, 361275, 481899, 1354828, 6328548, 6679881, ... (OEIS A005849; Caldwell). As of Nov. 2015, it is known that there are no other Cullen primes for  (PrimeGrid).

REFERENCES:

Caldwell, C. K. "The Top Twenty: Cullen Primes." https://primes.utm.edu/top20/page.php?id=6#records.

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. https://research.microsoft.com/~pleyland/factorization/cullen_woodall/cw.htm.

PrimeGrid. "PrimeGrid's Cullen Prime Search." Jul. 25, 2009. https://www.primegrid.com/download/Cullen6679881.pdf.

PrimeGrid. "Subprojects: Cullen Prime Search." https://www.primegrid.com/server_status_subprojects.php.

PrimeGrid. "PrimeGrid Primes: Subproject: (CUL) Cullen Prime Search." https://www.primegrid.com/primes/primes.php?project=CUL.

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

Rodenkirch, M. and Ballinger, R. "Cullen Primes: Definition and Status." https://www.prothsearch.net/cullen.html.

Sloane, N. J. A. Sequences A002064/M2795, A005849/M5401, and A050920 in "The On-Line Encyclopedia of Integer Sequences."