Powerful numbers (4,8,9,16,25,27... )are defined as follows: if a prime p divides n then p² 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!!

(In reply to An interesting remark ***** Spoiler by Ady TZIDON)

your discovery is easily proven
4n(n+1) is obviously also a p.n. be cause all of 4 n and n+1 are p.n.'s
now it remains to show that 4n(n+1) is also a p.n.
4n(n+1)+1
4n^2+4n+1
(2n+1)^2
now this is also a p.n. because for an prime divisor p of (2n+1), p^2 divides (2n+1)^2 thus all the prime divisor powers of (2n+1)^2 are at least 2

Posted by Daniel on 2010-08-09 18:16:30