# How many perfect powers are there amoung the first 1000 positive integers

A positive integer $$n$$ is called a perfect power if and only if there exist two positive integers $$p,q$$, both are greater than 1, such that $$n=p^q$$. For example $$8=2^3, 36=6^2$$ are perfect powers, but $$20=2^2*5, 11=11^1$$ are not.

Let $$A=\{1,2,\dots,1000\}$$. How many elements of $$A$$ is a perfect power?

I hope there exist a beautiful way to solve this, perhaps without calculating the number of primes untill 1000 or considering all the cases. Is there a formula? Thanks in advance!

I don't think there is anything better than counting up each power. There are $$\lfloor \sqrt {1000} \rfloor=31$$ squares and a similar formula for each power. You are correct that you only need to consider primes as all the fourth powers are squares as well. You also don't need to count any higher than $$10$$ because $$2^{10} \gt 1000$$ and you need to only count $$1$$ once. The sixth powers have been counted twice-once as squares and once as cubes, so need to be subtracted once. This would apply to any number the product of two primes, but six is the only one less than $$10$$.