Define, $\displaystyle \Phi(x,y)=\sum_{n\le x, \substack\\ p|n\implies p>y }1$. Prove that if $\sqrt x <y\le x$ then, $\Phi(x,y)=\pi(x)-\pi(y)+1$, where $\pi(x)$ denotes the number of primes less or equal to $x$.
I've deduced that $\displaystyle \Phi(x,y)=1+\sum_{y<p\le x}\Phi(x/p,p)$. Also we have, $\pi(x)-\pi(y)\le \Phi(x,y)$. From these two how to proceed further ?
You're actually looking in somewhat the wrong direction — rather, consider just breaking down the definition itself: for a number $n$ to "contribute to" $\Phi(x,y)$, by definition, it has to have all of its prime factors $\gt y$. Now, what can you say about composite numbers with all of their prime factors $\gt \sqrt{x}$?