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 "m^k", 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 = p₁^a₁p₂^a₂…p_k^a_k" 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 "m^k", 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