Cullen and Woodall numbers and their generalization to other bases

Cullen numbers are given by the expression n.2n + 1 and Woodall numbers by n.2n - 1. They are often abbreviated to C(n) and W(n) respectively. A generalization of Cullen and Woodall numbers is straightforward. The base 2 is replaced by another integer a, typically small, to define GC(a,n) as + 1 and GW(a,n) as - 1. Sometimes an even more abbreviated name turns out to be useful so on these pages a,n+ should be read as GC(a,n) and a,n- should be read as GW(a,n). Note that 2,n+ is the same as C(n) and 2,n- is just W(n).

The following tables all abide by my standard format and include factorizations for index n up to 1000.

(Generalized) Cullen

2+ 3+ 4+ 5+ 6+ 7+ 8+ 9+ 10+ 11+ 12+

(Generalized) Woodall

2- 3- 4- 5- 6- 7- 8- 9- 10- 11- 12-

The 10407 composite cofactors of these numbers are given in numerical order in this file. Each line consists of the decimal representation of a composite integer followed by a TAB followed by the abbreviated name for the number of which it is a factor. The data is rather large, currently almost 6 megabytes, and is gzipped for efficiency of storage and transmission.

Many people have helped to find factors of (G)CW numbers, sometimes unwittingly, and I have tried to give them appropriate credit. New discoveries are added to the progress tables as I learn of them. New contributions are always welcome and additional information is available for those wishing to help the GCW factorization project.

As of 2024-06-09, a total of 24739 factors have been discovered since I first made the tables public in September 2000 and a total of 11605 GCW numbers have been fully factored.

Some GCW numbers admit algebraic and/or Aurifeuillean factorizations. The algebraic.txt file gives further information.

Please contact me to report any errors in these pages or in the tables and for further information.

The tables are updated at sporadic intervals. The last update took place on 2024-06-09.