Polite number

[6][7][8][9][10][11][12] The problem of representing numbers as sums of consecutive integers and of counting the number of representations of this type has been studied by Sylvester,[13] Mason,[14][15] Leveque,[16] and many other more recent authors.

For every x, the politeness of x equals the number of odd divisors of x that are greater than one.

For instance, the polite number x = 14 has a single nontrivial odd divisor, 7.

It is therefore the sum of 7 consecutive numbers centered at 14/7 = 2: The first term, −1, cancels a later +1, and the second term, zero, can be omitted, leading to the polite representation Conversely, every polite representation of x can be formed from this construction.

[13][26] More generally, the same idea gives a two-to-one correspondence between, on the one hand, representations as a sum of consecutive integers (allowing zero, negative numbers, and single-term representations) and on the other hand odd divisors (including 1).

A polite representation has a single run, and a partition with one value d is equivalent to a factorization of n as the product d ⋅ (n/d), so the special case k = 1 of this result states again the equivalence between polite representations and odd factors (including in this case the trivial representation n = n and the trivial odd factor 1).

Thus, non-trapezoidal polite number must have the form of a power of two multiplied by an odd prime.

As Jones and Lord observe,[12] there are exactly two types of triangular numbers with this form: (sequence A068195 in the OEIS).