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.