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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Practical number ： ウィキペディア英語版 Practical number In number theory, a practical number or panarithmic number〔 cites and for the name "panarithmic numbers".〕 is a positive integer ''n'' such that all smaller positive integers can be represented as sums of distinct divisors of ''n''. For example, 12 is a practical number because all the numbers from 1 to 11 can be expressed as sums of its divisors 1, 2, 3, 4, and 6: as well as these divisors themselves, we have 5 = 3 + 2, 7 = 6 + 1, 8 = 6 + 2, 9 = 6 + 3, 10 = 6 + 3 + 1, and 11 = 6 + 3 + 2. The sequence of practical numbers begins :1, 2, 4, 6, 8, 12, 16, 18, 20, 24, 28, 30, 32, 36, 40, 42, 48, 54, 56, 60, 64, 66, 72, 78, 80, 84, 88, 90, 96, 100, 104, 108, 112, 120, 126, 128, 132, 140, 144, 150.... Practical numbers were used by Fibonacci in his Liber Abaci (1202) in connection with the problem of representing rational numbers as Egyptian fractions. Fibonacci does not formally define practical numbers, but he gives a table of Egyptian fraction expansions for fractions with practical denominators.〔.〕 The name "practical number" is due to , who first attempted a classification of these numbers that was completed by and . This characterization makes it possible to determine whether a number is practical by examining its prime factorization. Every even perfect number and every power of two is also a practical number. Practical numbers have also been shown to be analogous with prime numbers in many of their properties.〔; ; ; .〕 ==Characterization of practical numbers== As and showed, it is straightforward to determine whether a number is practical from its prime factorization. A positive integer $n=p\_1^...p\_k^$ with $n>1$ and primes :$p\_i\backslash leq1+\backslash sigma(p\_1^\backslash dots\; p\_^^\backslash frac,$ where $\backslash sigma(x)$ denotes the sum of the divisors of ''x''. For example, 3 ≤ σ(2)+1 = 4, 29 ≤ σ(2 × 32)+1 = 40, and 823 ≤ σ(2 × 32 × 29)+1=1171, so 2 × 32 × 29 × 823 = 429606 is practical. This characterization extends a partial classification of the practical numbers given by . It is not difficult to prove that this condition is necessary and sufficient for a number to be practical. In one direction, this condition is clearly necessary in order to be able to represent $p\_i-1$ as a sum of divisors of ''n''. In the other direction, the condition is sufficient, as can be shown by induction. More strongly, one can show that, if the factorization of ''n'' satisfies the condition above, then any $m\; \backslash le\; \backslash sigma(n)$ can be represented as a sum of divisors of ''n'', by the following sequence of steps: * Let $q\; =\; \backslash min\backslash )\backslash \}$, and let $r\; =\; m\; -\; qp\_k^$. * Since $q\backslash le\backslash sigma(n/p\_k^)$ and $n/p\_k^$ can be shown by induction to be practical, we can find a representation of ''q'' as a sum of divisors of $n/p\_k^$. * Since $r\backslash le\; \backslash sigma(n)\; -\; p\_k^\backslash sigma(n/p\_k^)\; =\; \backslash sigma(n/p\_k)$, and since $n/p\_k$ can be shown by induction to be practical, we can find a representation of ''r'' as a sum of divisors of $n/p\_k$. * The divisors representing ''r'', together with $p\_k^$ times each of the divisors representing ''q'', together form a representation of ''m'' as a sum of divisors of ''n''. 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「Practical number」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.