 Solution

Let $p_1,\dots ,p_a$ be the primes not exceeding $\sqrt{x}$, where $a=\pi (\sqrt{x})$, and let $\Phi (x,a)$ count integers $1\le n\le x$ divisible by none of these primes. Every surviving integer other than $1$ is prime: a composite integer at most $x$ has a prime factor at most its square root. Conversely, the surviving primes are exactly those between $\sqrt{x}$ and $x$. Thus the Legendre prime-counting formula isBy inclusion-exclusion,where the empty product is one. To cover literally every positive real $x$, replace the subtraction of $1$ by $1_{\{x\ge 1\}}$: for $0<x<1$, both counts are zero and there is no surviving integer $1$.

For $x=48$, the sieving primes are $2,3,5$. The surviving count is $48-(24+16+9)+(8+4+3)-1=13$. Consequently