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In mathematics, a highly abundant number is a natural number with the property that the sum of its divisors (including itself) is greater than the sum of the divisors of any smaller natural number. Highly abundant numbers and several similar classes of numbers were first introduced by , and early work on the subject was done by . Alaoglu and Erdős tabulated all highly abundant numbers up to 104, and showed that the number of highly abundant numbers less than any ''N'' is at least proportional to log2 ''N''. == Formal definition and examples == Formally, a natural number ''n'' is called highly abundant if and only if for all natural numbers ''m'' < ''n'', : where σ denotes the sum-of-divisors function. The first few highly abundant numbers are :1, 2, 3, 4, 6, 8, 12, 16, 18, 20, 24, 28, 30, 32, 36, 42, 48, 60, ... . For instance, 5 is not highly abundant because σ(5) = 5+1 = 6 is smaller than σ(4) = 4 + 2 + 1 = 7, while 8 is highly abundant because σ(8) = 8 + 4 + 2 + 1 = 15 is larger than all previous values of σ. The only odd highly abundant numbers are 1 and 3.〔See , p. 466. Alaoglu and Erdős claim more strongly that all highly abundant numbers greater than 210 are divisible by 4, but this is not true: 630 is highly abundant, and is not divisible by 4. (In fact, 630 is the only counterexample; all larger highly abundant numbers are divisible by 12.)〕 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「highly abundant number」の詳細全文を読む スポンサード リンク
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