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| An '''Achilles number''' is a number that is [[Powerful number|powerful]] but not a [[perfect power]].<ref name=mw>{{MathWorld|urlname=AchillesNumber|title=Achilles Number}}
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| </ref> A positive integer ''n'' is a powerful number if, for every [[prime factor]] ''p'' of ''n'', ''p''<sup>2</sup> 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 ''m<sup>k</sup>'', where ''m'' and ''k'' are positive integers greater than 1.
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| Achilles numbers are named after [[Achilles]], a hero of the [[Trojan war]], who was also powerful but imperfect.
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| ==Sequence of Achilles numbers==
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| A number ''n'' = ''p''<sub>1</sub><sup>''a''<sub>1</sub></sup>''p''<sub>2</sub><sup>''a''<sub>2</sub></sup>…''p''<sub>''k''</sub><sup>''a''<sub>''k''</sub></sup> is [[powerful number|powerful]] if min(''a''<sub>1</sub>, ''a''<sub>2</sub>, …, ''a''<sub>''k''</sub>) ≥ 2. If in addition gcd(''a''<sub>1</sub>, ''a''<sub>2</sub>, …, ''a''<sub>''k''</sub>) = 1 the number is an Achilles number.
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| The Achilles numbers up to 5000 are:
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| :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 {{OEIS|id=A052486}}.
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| The smallest pair of consecutive Achilles numbers is:<ref>Carlos Rivera, ''The Prime Puzzles and Problem Connection'', [http://www.primepuzzles.net/problems/prob_053.htm Problem 53]</ref>
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| : 5425069447 = 7<sup>3</sup> × 41<sup>2</sup> × 97<sup>2</sup>
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| : 5425069448 = 2<sup>3</sup> × 26041<sup>2</sup>
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| ==Examples==
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| 108 is a powerful number. Its [[prime factorization]] is 2<sup>2</sup> · 3<sup>3</sup>, and thus its prime factors are 2 and 3. Both 2<sup>2</sup> = 4 and 3<sup>2</sup> = 9 are divisors of 108. However, 108 cannot be represented as ''m<sup>k</sup>'', where ''m'' and ''k'' are positive integers greater than 1, so 108 is an Achilles number.
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| Finally, 784 is not an Achilles number. It is a powerful number, because not only are 2 and 7 its only prime factors, but also 2<sup>2</sup> = 4 and 7<sup>2</sup> = 49 are divisors of it. Nonetheless, it is a perfect power:
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| :<math>784=2^4 \cdot 7^2 = (2^2)^2 \cdot 7^2 = (2^2 \cdot 7)^2 = 28^2. \, </math>
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| So it is not an Achilles number.
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| == References ==
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| {{reflist}}
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| {{Divisor classes}}
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| {{Classes of natural numbers}}
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| {{DEFAULTSORT:Achilles Number}}
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| [[Category:Integer sequences]]
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