Smaller multiples were handled earlier. Every composite has a small prime factor, and earlier primes already mark smaller multiples.

Instead of testing each number independently, let each discovered prime announce all numbers it disproves. Begin where that announcement contains new information.

What the function promises: Return all primes <= limit. Limits below two return an empty list.

def primes_through(limit):
    if limit < 2: return []
    prime = bytearray(b"\x01") * (limit+1)
    prime[0:2] = b"\x00\x00"
    p = 2
    while p*p <= limit:
        if prime[p]:
            prime[p*p:limit+1:p] = b"\x00" * (((limit-p*p)//p)+1)
        p += 1
    return [i for i in range(2,limit+1) if prime[i]]

assert primes_through(20) == [2,3,5,7,11,13,17,19]
assert primes_through(1) == []

Work and storage: O(n log log n) time and O(n) bytes in the usual analysis.

When to recognize it: Every composite has a small prime factor, and earlier primes already mark smaller multiples.

Check before applying it: A full boolean array is unsuitable for enormous limits or narrow intervals far from zero.

Use primes through √R to mark a segmented sieve over that interval.