Let $K \ge 6$ (usually called twin index) be the number between a pair of twin primes, and let $k = K / 6$.
It is easy to see that all $k = n^2$ (where $n$ is a generic integer) are divisible by 5 (ex.: 25, 100,1225, 3600, 4900, 5625, 44100), but why are all numbers $k = n^3$ divisible by 7? Ex.: 21952, 74088, 4741632, 8365427, 23987543104.
It appears that there are no other numbers of the form $k = (a n)^b$ with "$a$" determined by "$b$" and both of them different from the pairs 5, 2 and 7, 3.
Can someone prove that?
