AnAchilles number is a number that ispowerful but not aperfect power.^[1] A positive integern is a powerful number if, for everyprime factorp ofn,p² is also adivisor. 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 asm^k, wherem andk are positive integers greater than 1.

Achilles numbers were named byHenry Bottomley afterAchilles, a hero of theTrojan War, who was also powerful but imperfect.Strong Achilles numbers are Achilles numbers whoseEuler totients are also Achilles numbers; the smallest are 500 and 864.^[2]

A numbern =p₁^a₁p₂^a₂…p_k^a_k ispowerful ifmin(a₁,a₂, …,a_k) ≥ 2. If in additiongcd(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 (sequenceA052486 in theOEIS).

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

5425069447 = 7³ × 41² × 97²

5425069448 = 2³ × 26041²

As an example, 108 is a powerful number. Itsprime 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 asm^k, wherem andk are positive integers greater than 1, so 108 is an Achilles number.

The integer 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. It is a perfect power:

${\displaystyle 784=2^4\cdot 7^2=(2^2{)}^2\cdot 7^2=(2^2\cdot 7{)}^2={28}^2.}$

So it is not an Achilles number.

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

• Integer sequences
• Achilles

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