python factorizaion function

fac.py

import time
from functools import wraps


def timeit(f):
    @wraps(f)
    def inner(*x, **y):
        a = time.monotonic()
        r = f(*x, **y)
        b = time.monotonic()
        print(f"elapsed: {b-a:.3f}s")
        return r

    return inner


@timeit
def factorization(n: int) -> list[tuple[int, int]]:
    res = []
    if n < 2:
        return res
    p = 2

    def helper(p: int) -> int:
        if p < 6:
            return (3, 5, 7)[(p - 1) // 2]
        t = 1 if p % 3 == 1 else 0
        l = int(n ** 0.5 + 0.5) + 1
        while p < l:
            p, t = p + (2, 4)[t], (t + 1) % 2
            if n % p == 0:
                return p
        return n

    while n > 1:
        e = 0
        while n % p == 0:
            n, e = n // p, e + 1
        if e > 0:
            res.append((p, e))
        p = helper(p)
    return res



@timeit
def fac(n: int) -> list[tuple[int, int]]:
    "enhanced version"
    res = []
    if n < 2:
        return res
    i = 1

    def helper(k: int) -> int:
        if k < 3:
            return k + 1
        if k < 8:
            return k + 2
        k = k + (4 if k % 3 == 1 else 2)
        l = int(n ** 0.5 + 1.5)
        while k < l:
            if n % k == 0:
                return k
            if n % (k + 2) == 0:
                return k + 2
            k += 6
        return n

    while n > 1:
        e = 0
        i = helper(i)
        while n % i == 0:
            n, e = n // i, e + 1
        if e > 0:
            res.append((i, e))
    return res


if __name__ == "__main__":
    ns = [3800651, 560411670, 10000128400406539]
    for n in ns:
        print(fac(n))
        print(factorization(n))