An Achilles number is a number that is powerful

Powerful number

A powerful number is a positive integer m such that for every prime number p dividing m, p2 also divides m. Equivalently, a powerful number is the product of a square and a cube, that is, a number m of the form m = a2b3, where a and b are positive integers. Powerful numbers are also known as...

 but not a perfect power. A positive integer n is a powerful number if, for every prime divisor

Divisor

In mathematics, a divisor of an integer n, also called a factor of n, is an integer which divides n without leaving a remainder.-Explanation:...

 or factor p of n, p² 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 are named after Achilles

Achilles

In Greek mythology, Achilles was a Greek hero of the Trojan War, the central character and the greatest warrior of Homer's Iliad.Plato named Achilles the handsomest of the heroes assembled against Troy....

, a hero of the Trojan war

Trojan War

In Greek mythology, the Trojan War was waged against the city of Troy by the Achaeans after Paris of Troy took Helen from her husband Menelaus, the king of Sparta. The war is among the most important events in Greek mythology and was narrated in many works of Greek literature, including the Iliad...

, who was also powerful but imperfect.

Sequence of Achilles numbers

A number n = p₁^a₁p₂^a₂…p_k^a_k is powerful

Powerful number

A powerful number is a positive integer m such that for every prime number p dividing m, p2 also divides m. Equivalently, a powerful number is the product of a square and a cube, that is, a number m of the form m = a2b3, where a and b are positive integers. Powerful numbers are also known as...

 if min(a₁, a₂, …, a_k) ≥ 2. If in addition gcd(a₁, a₂, …, 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 .

The smallest pair of consecutive Achilles numbers is:

5425069447 = 7³ × 41² × 97²

5425069448 = 2³ × 26041²

Examples

108 is a powerful number. Its prime factorization is 2² · 3³, 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:


So it is not an Achilles number.