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In mathematics, a colossally abundant number (sometimes abbreviated as CA) is a natural number that, in a particular, rigorous sense, has many divisors. Formally, a number ''n'' is colossally abundant if and only if there is an ε > 0 such that for all ''k'' > 1, : where σ denotes the sum-of-divisors function.〔K. Briggs, "Abundant Numbers and the Riemann Hypothesis", ''Experimental Mathematics'' 15:2 (2006), pp. 251–256, .〕 All colossally abundant numbers are also superabundant numbers, but the converse is not true. The first 15 colossally abundant numbers, 2, 6, 12, 60, 120, 360, 2520, 5040, 55440, 720720, 1441440, 4324320, 21621600, 367567200, 6983776800 are also the first 15 superior highly composite numbers. ==History== Colossally abundant numbers were first studied by Ramanujan and his findings were intended to be included in his 1915 paper on were mostly conditional on the Riemann hypothesis and with this assumption he found upper and lower bounds for the size of colossally abundant numbers and proved that what would come to be known as Robin's inequality (see below) holds for all sufficiently large values of ''n''.〔S. Ramanujan, "Highly composite numbers. Annotated and with a foreword by J.-L. Nicholas and G. Robin", ''Ramanujan Journal'' 1 (1997), pp. 119–153.〕 The class of numbers was reconsidered in a slightly stronger form in a 1944 paper of Leonidas Alaoglu and Paul Erdős in which they tried to extend Ramanujan's results.〔citation | last1 = Alaoglu | first1 = L. | author1-link = Leonidas Alaoglu | last2 = Erdös | first2 = P. | author2-link = Paul Erdős | journal = Transactions of the American Mathematical Society | mr = 0011087 | pages = 448–469 | title = On highly composite and similar numbers | url = http://www.renyi.hu/~p_erdos/1944-03.pdf | volume = 56 | year = 1944}}.〕 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「Colossally abundant number」の詳細全文を読む スポンサード リンク
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