polite number

A polite number $n$ is an integer that is the sum of two or more consecutive nonnegative integers in at least one way. To put it algebraically, if $n$ is polite then there is a solution to

$$n=\sum _{i=a}^{b}i$$

with $b>a$ and $a>-1$. For example, 42 is a polite number since it is the sum of the integers from 3 to 9. The first few polite numbers are 3, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 40, etc.

Obviously all triangular numbers are polite numbers. So are all odd numbers. In fact, the numbers that are not polite are the powers of 2.

Title	polite number
Canonical name	PoliteNumber
Date of creation	2013-03-22 18:09:54
Last modified on	2013-03-22 18:09:54
Owner	PrimeFan (13766)
Last modified by	PrimeFan (13766)
Numerical id	6
Author	PrimeFan (13766)
Entry type	Definition
Classification	msc 11A25