 Smallest prime factor sieve

A smallest prime factor sieve stores the least prime divisor $\mathrm{spf}(n)$ for integers $n\ge 2$ up to a bound $N$. Initialize entries unset; scan primes in increasing order and fill their still-unset multiples. A composite $n$ first marked by $p$ has no smaller prime divisor, so the stored value is correct. For example, repeatedly dividing $84$ by its stored factors gives $84\to 42\to 21\to 7\to 1$, hence $84=2^2\cdot 3\cdot 7$. A table uses $O(N)$ entries and accelerates repeated factorizations; the values at zero and one need a separate sentinel convention.