Polite 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. A positive integer which is not polite is called impolite.[1][2] The impolite numbers are exactly the powers of two, and the polite numbers are the natural numbers that are not powers of two.
Polite numbers have also been called staircase numbers because the Young diagrams which represent graphically the partitions of a polite number into consecutive integers (in the French notation of drawing these diagrams) resemble staircases.[3][4][5] 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.[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.[1][2][17][18][19][20][21][22][23] The polite numbers describe the possible numbers of sides of the Reinhardt polygons.[24]