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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 Powerful number ： ウィキペディア英語版 Powerful number A powerful number is a positive integer ''m'' such that for every prime number ''p'' dividing ''m'', ''p''2 also divides ''m''. Equivalently, a powerful number is the product of a square and a cube, that is, a number ''m'' of the form ''m'' = ''a''2''b''3, where ''a'' and ''b'' are positive integers. Powerful numbers are also known as squareful, square-full, or 2-full. Paul Erdős and George Szekeres studied such numbers and Solomon W. Golomb named such numbers ''powerful''. The following is a list of all powerful numbers between 1 and 1000: :1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 72, 81, 100, 108, 121, 125, 128, 144, 169, 196, 200, 216, 225, 243, 256, 288, 289, 324, 343, 361, 392, 400, 432, 441, 484, 500, 512, 529, 576, 625, 648, 675, 676, 729, 784, 800, 841, 864, 900, 961, 968, 972, 1000, ... . == Equivalence of the two definitions == If ''m'' = ''a''2''b''3, then every prime in the prime factorization of ''a'' appears in the prime factorization of ''m'' with an exponent of at least two, and every prime in the prime factorization of ''b'' appears in the prime factorization of ''m'' with an exponent of at least three; therefore, ''m'' is powerful. In the other direction, suppose that ''m'' is powerful, with prime factorization :$m\; =\; \backslash prod\; p\_i^,$ where each αi ≥ 2. Define γi to be three if αi is odd, and zero otherwise, and define βi = αi - γi. Then, all values βi are nonnegative even integers, and all values γi are either zero or three, so :$m\; =\; (\backslash prod\; p\_i^)(\backslash prod\; p\_i^)\; =\; (\backslash prod\; p\_i^)^2(\backslash prod\; p\_i^)^3$ supplies the desired representation of ''m'' as a product of a square and a cube. Informally, given the prime factorization of ''m'', take ''b'' to be the product of the prime factors of ''m'' that have an odd exponent (if there are none, then take ''b'' to be 1). Because ''m'' is powerful, each prime factor with an odd exponent has an exponent that is at least 3, so ''m''/''b''3 is an integer. In addition, each prime factor of ''m''/''b''3 has an even exponent, so ''m''/''b''3 is a perfect square, so call this ''a''2; then ''m'' = ''a''2''b''3. For example: :$m\; =\; 21600\; =\; 2^5\; \backslash times\; 3^3\; \backslash times\; 5^2\; \backslash ,\; ,$ :$b\; =\; 2\; \backslash times\; 3\; =\; 6\; \backslash ,\; ,$ :$a\; =\; \backslash sqrt\}\; =\; \backslash sqrt\; =\; 10\; \backslash ,\; ,$ :$m\; =\; a^2b^3\; =\; 10^2\; \backslash times\; 6^3\; \backslash ,\; .$ The representation ''m'' = ''a''2''b''3 calculated in this way has the property that ''b'' is squarefree, and is uniquely defined by this property. 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「Powerful 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.