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In mathematics, a multiply perfect number (also called ''multiperfect number'' or ''pluperfect number'') is a generalization of a perfect number. For a given natural number ''k'', a number ''n'' is called ''k''-perfect (or ''k''-fold perfect) if and only if the sum of all positive divisors of n (the divisor function, ''σ(n)'') is equal to ''kn''; a number is thus perfect if and only if it is 2-perfect. A number that is ''k''-perfect for a certain ''k'' is called a multiply perfect number. As of 2014, ''k''-perfect numbers are known for each value of ''k'' up to 11.〔 It can be proven that: * For a given prime number ''p'', if ''n'' is ''p''-perfect and ''p'' does not divide ''n'', then ''pn'' is (''p''+1)-perfect. This implies that an integer ''n'' is a 3-perfect number divisible by 2 but not by 4, if and only if ''n''/2 is an odd perfect number, of which none are known. * If 3''n'' is 4''k''-perfect and 3 does not divide ''n'', then n is 3''k''-perfect. == Smallest ''k''-perfect numbers == The following table gives an overview of the smallest ''k''-perfect numbers for ''k'' ≤ 11 : For example, 120 is 3-perfect because the sum of the divisors of 120 is 1+2+3+4+5+6+8+10+12+15+20+24+30+40+60+120 = 360 = 3 × 120. 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「multiply perfect number」の詳細全文を読む スポンサード リンク
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