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 factorp 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.

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}^{a1}p_{2}^{a2}…p_{k}^{ak} is powerful if min(a_{1}, a_{2}, …, a_{k}) ≥ 2. If in addition gcd(a_{1}, a_{2}, …, a_{k}) = 1 the number is an Achilles number.

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

5425069447 = 7^{3} × 41^{2} × 97^{2}

5425069448 = 2^{3} × 26041^{2}

Examples

108 is a powerful number. Its prime factorization is 2^{2} · 3^{3}, and thus its prime factors are 2 and 3. Both 2^{2} = 4 and 3^{2} = 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 5^{2} = 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 2^{2} = 4 and 7^{2} = 49 are divisors of it. It is a perfect power: