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In mathematics, a superperfect number is a positive integer ''n'' that satisfies : where σ is the divisor function. Superperfect numbers are a generalization of perfect numbers. The term was coined by Suryanarayana (1969).〔 The first few superperfect numbers are :2, 4, 16, 64, 4096, 65536, 262144, 1073741824, ... . If ''n'' is an ''even'' superperfect number then ''n'' must be a power of 2, 2''k'', such that 2''k''+1-1 is a Mersenne prime.〔 It is not known whether there are any ''odd'' superperfect numbers. An odd superperfect number ''n'' would have to be a square number such that either ''n'' or σ(''n'') is divisible by at least three distinct primes.〔 There are no odd superperfect numbers below 7x1024.〔Guy (2004) p.99〕 ==Generalisations== Perfect and superperfect numbers are examples of the wider class of ''m''-superperfect numbers, which satisfy : corresponding to ''m''=1 and 2 respectively. For ''m'' ≥ 3 there are no even ''m''-superperfect numbers.〔 The ''m''-superperfect numbers are in turn examples of (''m'',''k'')-perfect numbers which satisfy〔Cohen & te Riele (1996)〕 : With this notation, perfect numbers are (1,2)-perfect, multiperfect numbers are (1,''k'')-perfect, superperfect numbers are (2,2)-perfect and ''m''-superperfect numbers are (''m'',2)-perfect.〔Guy (2007) p.79〕 Examples of classes of (''m'',''k'')-perfect numbers are: : 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「superperfect number」の詳細全文を読む スポンサード リンク
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