Numbers with special prime factorization

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.

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]}

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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 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 = 7
^{3} × 41^{2} × 97^{2}
- 5425069448 = 2
^{3} × 26041^{2}

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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. 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 = 2^{2} × 5^{3} is a strong Achilles number as its Euler totient of 200 = 2^{3} × 5^{2} is also an Achilles number.