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An Achilles number is a number that is powerful but not a perfect power.〔 〕 A positive integer is a powerful number if, for every prime factor of , is also a divisor. In other words, every prime factor appears at least squared in the factorization. All Achilles numbers are powerful. However, not all powerful numbers are Achilles numbers: only those that cannot be represented as , where and are positive integers greater than 1. Achilles numbers are named after Achilles, a hero of the Trojan war, who was also powerful but imperfect. ==Sequence of Achilles numbers== A number is powerful if . If in addition the number is an Achilles number. The Achilles numbers up to 5000 are: :72, 108, 200, 288, 392, 432, 500, 648, 675, 800, 864, 968, 972, 1125, 1152, 1323, 1352, 1372, 1568, 1800, 1944, 2000, 2312, 2592, 2700, 2888, 3087, 3200, 3267, 3456, 3528, 3872, 3888, 4000, 4232, 4500, 4563, 4608, 5000 . The smallest pair of consecutive Achilles numbers is:〔Carlos Rivera, ''The Prime Puzzles and Problem Connection'', (Problem 53 )〕 : 5425069447 = 73 × 412 × 972 : 5425069448 = 23 × 260412 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「Achilles number」の詳細全文を読む スポンサード リンク
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