Finding the number of square prefixes of a string in linear time
Suppose there is an $O(|W|)$ algorithm that computes a slightly different prefix function: $Z[i]$ is the longest prefix of $W$ that is also a prefix of $W[i..n]$. Note then that your answer will simply be the count of indices $i$ such that $Z[i] &gt;= i$, assuming 0-indexing. 

Fortunately, such an algorithm exists and is quite elegant! It's (as always) a good idea to try to devise it on your own. If you get stuck, here's a pretty good explanation: http://codeforces.com/blog/entry/3107.