Powerful numbers (4,8,9,16,25,27... )are defined as follows: if a prime p divides n then p^{2} must also divide n.
(8,9) is a couple of two consecutive numbers,both of them being powerful.
Find another pair(s) like that.
The more the merrier!!
According to the definition, 36 is a powerful number. (2 and 3 are the only primes which divide it, and 2^2 and 3^2 divide as well.) I think it's just because the first few powerful numbers are perfect powers, that these weren't included in the examples.
However note numbers like n=20 are not powerful by that definition, as 5 is a prime which divides n, but p^2 does not divide n. The definition "If a prime factor does this, it must also do this" is actually a statement over all prime factors. However of course, most (almost all) of them do not divide n, and so do not apply.

Posted by Gamer
on 20100809 13:39:36 