
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.〔.〕〔.〕〔.〕〔.〕〔.〕〔.〕〔.〕 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,〔. In (The collected mathematical papers of James Joseph Sylvester (December 1904) ), H. F. Baker, ed. Sylvester defines the ''class'' of a partition into distinct integers as the number of blocks of consecutive integers in the partition, so in his notation a polite partition is of first class.〕 Mason,〔.〕〔.〕 Leveque,〔,〕 and many other more recent authors.〔〔〔.〕〔.〕〔.〕〔.〕〔.〕〔.〕〔.〕 ==Examples and characterization== 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, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, ... . The impolite numbers are exactly the powers of two.〔 It follows from the Lambek–Moser theorem that the ''n''th polite number is ƒ(''n'' + 1), where :$f(n)=n+\backslash left\backslash lfloor\backslash log\_2\backslash left(n+\backslash log\_2\; n\backslash right)\backslash right\backslash rfloor.$ 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Polite number」の詳細全文を読む スポンサード リンク
