For example, , and all the triangular numbers greater than 1 are polite numbers (I do not allow .)
It turns out that almost all the numbers are polite, and the only impolite numbers are the powers of 2.
Indeed, a number with odd divisors greater than 1 can be represented in ways as a nontrivial sum of consecutive naturals.
For example, has 3 odd divisors greater than 1, namely , and itself, and 3 representations
Since polite numbers are so common, I prefer to list impolite numbers.