The old version fully marked the entire primes table before yielding its first value. Now subsequences of primes are yielded as soon as the subsequence becomes fully marked. This improves responsiveness — the first call to next() returns more quickly. And since the work is done lazily, the factor() recipe saves work when small factors reduce the search space. For example factor(12**20) is now several times faster.

The old version did some perplexing manipulations to the data array, data[:3] = 0, 0, 0 and data[2] = 1. The new version is straight forward and has clean loop invariants:

• data[start : p*p] has valid primes that have not yet been yielded
• data[p*p : n] still needs to be marked for multiples of p.

The new version is easy experiment with and can be easily changedto the simplest version of the algorithm that isn't lazy and hasn't optimized away the even composites.

def sieve(n):
       start = 2
       data = bytearray([1]) * n
       limit = math.isqrt(n) + 1
       for p in iter_index(data, 1, start, limit):
           data[p*p : n : p] = bytes(len(range(p*p, n, p)))
       yield from iter_index(data, 1, start)

📚 Documentation preview 📚: https://cpython-previews--109199.org.readthedocs.build/