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 factor $p$ of $n$ , $p 2$ is also a divisor. In other words, every prime factor appears at least squared in the factorization. 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.

Demonstration, with Cuisenaire rods, of the number 72 being powerful

Achilles numbers were named by Henry Bottomley after Achilles, a hero of the Trojan war, who was also powerful but imperfect. Strong Achilles numbers are Achilles numbers whose Euler totients are also Achilles numbers.[2]

Sequence of Achilles numbers

A number $n = p 1 a 1 p 2 a 2 \dots p k a k$ is powerful if $min(a 1, a 2, \dots, a k) \geq 2$ . If in addition $gcd(a 1, a 2, \dots, a k) = 1$ 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 the OEIS).

The smallest pair of consecutive Achilles numbers is:[3]

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.

360 is not an Achilles number because it is not powerful. One of its prime factors is 5 but 360 is not divisible by 5² = 25.

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.

500 = 2² × 5³ is a strong Achilles number as its Euler totient of 200 = 2³ × 5² is also an Achilles number.

References

Weisstein, Eric W. "Achilles Number". MathWorld.
"Problem 302 - Project Euler". projecteuler.net.
Carlos Rivera, The Prime Puzzles and Problem Connection, Problem 53

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[mw-1] Weisstein, Eric W. "Achilles Number". MathWorld.

[2] "Problem 302 - Project Euler". projecteuler.net.

[3] Carlos Rivera, The Prime Puzzles and Problem Connection, Problem 53

Divisibility-based sets of integers
Overview	Integer factorization Divisor Unitary divisor Divisor function Prime factor Fundamental theorem of arithmetic
Factorization forms	Prime Composite Semiprime Pronic Sphenic Square-free Powerful Perfect power Achilles Smooth Regular Rough Unusual
Constrained divisor sums	Perfect Almost perfect Quasiperfect Multiply perfect Hemiperfect Hyperperfect Superperfect Unitary perfect Semiperfect Practical Erdős–Nicolas
With many divisors	Abundant Primitive abundant Highly abundant Superabundant Colossally abundant Highly composite Superior highly composite Weird
Aliquot sequence-related	Untouchable Amicable Sociable Betrothed
Base-dependent	Equidigital Extravagant Frugal Harshad Polydivisible Smith
Other sets	Arithmetic Deficient Friendly Solitary Sublime Harmonic divisor Descartes Refactorable Superperfect