Trapezoidal number
In number theory, a polite number is a positive integer that can be written as the sum of two or more consecutive positive integers. Other positive integers are impolite. Polite numbers have also been called staircase numbers because the Young diagrams representing graphically the partitions of a polite number into consecutive integers (in the French style of drawing these diagrams) resemble staircases. If all numbers in the sum are strictly greater than one, the numbers so formed are also called trapezoidal numbers because they represent patterns of points arranged in a trapezoid (trapezium outside North America).
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, Mason,Leveque, and many other more recent authors.
The first few polite numbers are
The impolite numbers are exactly the powers of two. It follows from the Lambek–Moser theorem that the nth polite number is f(n + 1), where
The politeness of a positive number is defined as the number of ways it can be expressed as the sum of consecutive integers. For every x, the politeness of x equals the number of odd divisors of x that are greater than one. The politeness of the numbers 1, 2, 3, ... is
For instance, the politeness of 9 is 2 because it has two odd divisors, 3 and itself, and two polite representations
the politeness of 15 is 3 because it has three odd divisors, 3, 5, and 15, and (as is familiar to cribbage players) three polite representations
An easy way of calculating the politeness of a positive number is that of decomposing the number into its prime factors, taking the powers of all prime factors greater than 2, adding 1 to all of them, multiplying the numbers thus obtained with each other and subtracting 1. For instance 90 has politeness 5 because ; the powers of 3 and 5 are respectively 2 and 1, and applying this method .
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