Achilles number
From Wikipedia, the free encyclopedia
| Divisibility-based sets of integers |
| Form of factorization: |
| Prime number |
| Composite number |
| Powerful number |
| Square-free number |
| Achilles number |
| Constrained divisor sums: |
| Perfect number |
| Almost perfect number |
| Quasiperfect number |
| Multiply perfect number |
| Hyperperfect number |
| Unitary perfect number |
| Semiperfect number |
| Primitive semiperfect number |
| Practical number |
| Numbers with many divisors: |
| Abundant number |
| Highly abundant number |
| Superabundant number |
| Colossally abundant number |
| Highly composite number |
| Superior highly composite number |
| Other: |
| Deficient number |
| Weird number |
| Amicable number |
| Sociable number |
| Sublime number |
| Harmonic divisor number |
| Frugal number |
| Equidigital number |
| Extravagant number |
| See also: |
| Divisor function |
| Divisor |
| Prime factor |
| Factorization |
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, p2 is also a divisor. In other words, every prime factor appears 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 a not a perfect power): not entirely unlike Achilles, a hero of the Trojan war.
[edit] Sequence of Achilles numbers
The Achilles numbers between 1 and 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)
[edit] Examples
108 is a powerful number. Its prime factorization is 
, 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 are 22 = 4 and 72 = 49 divisors of it. Nonetheless, it is a perfect power:
So it is not an Achilles number.
