43 lines
2.4 KiB
Text
43 lines
2.4 KiB
Text
An '''Achilles number''' is a number that is powerful but imperfect. ''Named after Achilles, a hero of the Trojan war, who was also powerful but imperfect.''
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A positive integer '''n''' is a powerful number if, for every prime factor '''p''' of '''n''', '''p<sup>2</sup>''' is also a divisor.
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In other words, every prime factor appears at least squared in the factorization.
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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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A '''''strong'' Achilles number''' is an '''Achilles number''' whose '''Euler totient (𝜑)''' is also an '''Achilles number'''.
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;E.G.
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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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'''360''' is ''not'' an '''Achilles number''' because it is not powerful. One of its prime factors is '''5''' but '''360''' is not divisible by '''5<sup>2</sup> = 25'''.
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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; its square root is an even integer, so it is ''not'' an '''Achilles number'''.
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'''500 = 2<sup>2</sup> × 5<sup>3</sup>''' is a '''''strong'' Achilles number''' as its Euler totient, '''𝜑(500)''', is '''200 = 2<sup>3</sup> × 5<sup>2</sup>''' which is ''also'' an '''Achilles number'''.
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;Task
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* Find and show the first 50 '''Achilles numbers'''.
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* Find and show at least the first 20 '''''strong'' Achilles numbers'''.
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* For at least 2 through 5, show the count of '''Achilles numbers''' with that many digits.
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;See also
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;* [[wp:Achilles number|Wikipedia: Achilles number]]
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;* [[oeis:A052486|OEIS:A052486 - Achilles numbers - powerful but imperfect numbers]]
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;* [[oeis:A194085|OEIS:A194085 - Strong Achilles numbers: Achilles numbers m such that phi(m) is also an Achilles number]]
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;* [[Powerful numbers|Related task: Powerful numbers]]
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;* [[Totient function|Related task: Totient function]]
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