Achilles number

Achilles number

An Achilles number is a number that is powerful but not a perfect power. A positive integer "n" is a powerful number if, for every prime divisor or factor "p" of "n", "p"2 is also a divisor. In other words, every prime factor appears at least squared. All Achilles numbers are powerful. However, not all powerful numbers are Achilles numbers: only those that cannot be represented as "mk", where "m" and "k" are positive integers greater than 1.

Achilles numbers, put laconically, are powerful but imperfect (as in not a perfect power) like Achilles, a hero of the Trojan war.

equence of Achilles numbers

A number "n = p1a1p2a2…pkak" is powerful if min("a"1, "a"2, …, "a""k") ≥ 2. If in addition gcd("a"1, "a"2, …, "a""k") = 1 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 OEIS|id=A052486.

Examples

108 is a powerful number. Its prime factorization is 2^2 cdot 3^3, and thus its prime factors are 2 and 3. Both 22 = 4 and 32 = 9 are divisors of 108. However, 108 cannot be represented as "mk", where "m" and "k" are positive integers greater than 1, so 108 is an Achilles number.

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 22 = 4 and 72 = 49 are divisors of it. Nonetheless, it is a perfect power::784=2^4 cdot 7^2 = (2^2)^2 cdot 7^2 = (2^2 cdot 7)^2 = 28^2.So it is not an Achilles number.

External links

*MathWorld|urlname=AchillesNumber|title=Achilles Number


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