# Definition:Cullen Number

## Definition

A Cullen number is a positive integer of the form:

$n \times 2^n + 1$

### Sequence

The sequence of Cullen numbers begins:

$1, 3, 9, 25, 65, 161, 385, \ldots$

corresponding to $n = 0, 1, 2, 3, \ldots$

## Also defined as

Some sources refer to numbers of the form $n \times 2^n - 1$ as also being Cullen numbers.

However, it is now conventional to refer to numbers of the form $n \times 2^n - 1$ as Woodall numbers, for Herbert J. Woodall.

## Also known as

Some sources refer to Cullen numbers as Cunningham numbers, for Allan Joseph Champneys Cunningham, so as to ensure their distinction from Woodall numbers.

## Source of Name

This entry was named for James Cullen.