Achilles number

An Achilles number is a number that is powerful but not a perfect power.^[1] A positive integer n is a powerful number if, for every prime divisor 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, a hero of the Trojan war, who was also powerful but imperfect.

Sequence of Achilles numbers

A number n = p₁^a₁p₂^a₂…p_k^a_k is powerful 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 (sequence A052486 in OEIS).

The smallest pair of consecutive Achilles numbers is:^[2]

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.

References

v t e Divisibility-based sets of integers Overview Integer factorization Divisor Unitary divisor Divisor function Prime factor Fundamental theorem of arithmetic Forms of factorization Prime number Composite number Semiprime number Pronic number Sphenic number Square-free number Powerful number Perfect power Achilles number Smooth number Regular number Rough number Unusual number Constrained divisor sums Perfect number Almost perfect number Quasiperfect number Multiply perfect number Hyperperfect number Superperfect number Unitary perfect number Semiperfect number Primitive semiperfect number Practical number With many divisors Abundant number Primitive abundant number Highly abundant number Superabundant number Colossally abundant number Highly composite number Superior highly composite number Weird number Related to aliquot sequences Untouchable number Amicable number Sociable number Other sets Deficient number Friendly number Solitary number Sublime number Harmonic divisor number Frugal number Equidigital number Extravagant number