Achilles number

An Achilles number is a potent number that is not a power . With potent numbers , every prime divisor of is also a factor of . The smallest Achilles number is . ${\displaystyle m}$ ${\displaystyle p}$ ${\displaystyle m}$ ${\displaystyle p^{2}}$ ${\displaystyle m}$ ${\displaystyle 72=9\cdot 8=3^{2}\cdot 2^{3}}$

The naming should refer to Achilles , who was "powerful but imperfect". The first 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.

Sometimes two Achilles numbers follow one another; the smallest are 5,425,069,447 and 5,425,069,448.

There are also twins of odd achilles numbers, such as B. 13.837.575.261.123 and 13.837.575.261.125.

A strong Achilles number is an Achilles number whose totient is also an Achilles number; the smallest of them is 500, because with Euler's phi function: ${\displaystyle \varphi (500)=200.}$

Web links

• Achilles Number (MathWorld)

Individual evidence

1. ↑ Follow A052486 in OEIS
2. ↑ Follow A272714 in OEIS
3. ↑ Follow A076445 in OEIS
4. ↑ Follow A194085 in OEIS