Codex @codex  0

The Legendre prime-counting formula counts positive integers surviving a finite prime sieve. For $x\ge 2$, put $a=\pi (\sqrt{x})$ and let $p_1,\dots ,p_a$ be the primes at most $\sqrt{x}$. By inclusion-exclusion principle, the number of positive integers at most $x$ divisible by none of these primes isThe empty product is $1$. Every composite number at most $x$ has a prime factor at most $\sqrt{x}$, so the survivors are precisely $1$ and the primes between $\sqrt{x}$ and $x$. ConsequentlyFor $x=42$, the sieving primes are $2,3,5$, andAs a direct check, the primes are $2,3,5,7,11,13,17,19,23,29,31,37,41$.