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A unitary perfect number is an integer which is the sum of its positive proper unitary divisors, not including the number itself. (A divisor ''d'' of a number ''n'' is a unitary divisor if ''d'' and ''n''/''d'' share no common factors.) Some perfect numbers are not unitary perfect numbers, and some unitary perfect numbers are not regular perfect numbers. Thus, 60 is a unitary perfect number, because 1, 3, 4, 5, 12, 15 and 20 are its proper unitary divisors, and 1 + 3 + 4 + 5 + 12 + 15 + 20 = 60. The first five, and only known, unitary perfect numbers are: 6, 60, 90, 87360, 146361946186458562560000 The respective sums of proper unitary divisors: * 6 = 1 + 2 + 3 * 60 = 1 + 3 + 4 + 5 + 12 + 15 + 20 * 90 = 1 + 2 + 5 + 9 + 10 + 18 + 45 * 87360 = 1 + 3 + 5 + 7 + 13 + 15 + 21 + 35 + 39 + 64 + 65 + 91 + 105 + 192 + 195 + 273 + 320 + 448 + 455 + 832 + 960 + 1344 + 1365 + 2240 + 2496 + 4160 + 5824 + 6720 + 12480 + 17472 + 29120 * 146361946186458562560000 = 1 + 3 + 7 + 11 + ... 13305631471496232960000 + 20908849455208366080000 + 48787315395486187520000 (4095 divisors in the sum) There are no odd unitary perfect numbers. This follows since one has 2''d'' *(''n'') dividing the sum of the unitary divisors of an odd number (where ''d'' *(''n'') is the number of distinct prime divisors of n). One gets this because the sum of all the unitary divisors is a multiplicative function and one has the sum of the unitary divisors of a power of a prime ''p''''a'' is ''p''''a'' + 1 which is even for all odd primes ''p''. Therefore, an odd unitary perfect number must have only one distinct prime factor, and it is not hard to show that a power of prime cannot be a unitary perfect number, since there are not enough divisors. It is not known whether or not there are infinitely many unitary perfect numbers, or indeed whether there are any further examples beyond the five already known. A sixth such number would have at least nine odd prime factors. == References == * Section B3. * * * 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「Unitary perfect number」の詳細全文を読む スポンサード リンク
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