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A centered nonagonal number is a centered figurate number that represents a nonagon with a dot in the center and all other dots surrounding the center dot in successive nonagonal layers. The centered nonagonal number for ''n'' is given by the formula : Multiplying the (''n'' - 1)th triangular number by 9 and then adding 1 yields the ''n''th centered nonagonal number, but centered nonagonal numbers have an even simpler relation to triangular numbers: every third triangular number (the 1st, 4th, 7th, etc.) is also a centered nonagonal number.〔 Thus, the first few centered nonagonal numbers are〔 :1, 10, 28, 55, 91, 136, 190, 253, 325, 406, 496, 595, 703, 820, 946. The list above includes the perfect numbers 28 and 496. All even perfect numbers are triangular numbers whose index is an odd Mersenne prime.〔.〕 Since every Mersenne prime greater than 3 is congruent to 1 modulo 3, it follows that every even perfect number greater than 6 is a centered nonagonal number. In 1850, Sir Frederick Pollock conjectured that every natural number is the sum of at most eleven centered nonagonal numbers, which has been neither proven nor disproven.〔.〕 ==See also== *Nonagonal number 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「Centered nonagonal number」の詳細全文を読む スポンサード リンク
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