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In mathematics, a harmonic divisor number, or Ore number (named after Øystein Ore who defined it in 1948), is a positive integer whose divisors have a harmonic mean that is an integer. The first few harmonic divisor numbers are :1, 6, 28, 140, 270, 496, 672, 1638, 2970, 6200, 8128, 8190 . For example, the harmonic divisor number 6 has the four divisors 1, 2, 3, and 6. Their harmonic mean is an integer: : The number 140 has divisors 1, 2, 4, 5, 7, 10, 14, 20, 28, 35, 70, and 140. Their harmonic mean is: : 5 is an integer, making 140 a harmonic divisor number. == Harmonic divisor numbers and perfect numbers == For any integer ''M'', as Ore observed, the product of the harmonic mean and arithmetic mean of its divisors equals ''M'' itself, as can be seen from the definitions. Therefore, ''M'' is harmonic, with harmonic mean of divisors ''k'', if and only if the average of its divisors is the product of ''M'' with a unit fraction 1/''k''. Ore showed that every perfect number is harmonic. To see this, observe that the sum of the divisors of a perfect number ''M'' is exactly ''2M''; therefore, the average of the divisors is ''M''(2/τ(''M'')), where τ(''M'') denotes the number of divisors of ''M''. For any ''M'', τ(''M'') is odd if and only if ''M'' is a square number, for otherwise each divisor ''d'' of ''M'' can be paired with a different divisor ''M''/''d''. But, no perfect number can be a square: this follows from the known form of even perfect numbers and from the fact that odd perfect numbers (if they exist) must have a factor of the form ''q''α where α ≡ 1 (mod 4). Therefore, for a perfect number ''M'', τ(''M'') is even and the average of the divisors is the product of ''M'' with the unit fraction 2/τ(''M''); thus, ''M'' is a harmonic divisor number. Ore conjectured that no odd harmonic divisor numbers exist other than 1. If the conjecture is true, this would imply the nonexistence of odd perfect numbers. 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「Harmonic divisor number」の詳細全文を読む スポンサード リンク
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