def s_for_prime_power(p, e):
    k = 0
 
    while e > p:
        k += p
        e -= p + 1
        t = k // p
 
        while t % p == 0:
            t //= p
            e -= 1
 
    return (k + max(0, e)) * p
 
def div_of_factorials(start, limit):
    small_prime_limit = limit ** 0.5
    small_primes = []
    half = limit // 2
 
    s = [0] * (limit + 1)
    m = 2
 
    for m in range(2, half + 1):
        if s[m] == 0:
            s[m] = m
 
            if m <= small_prime_limit:
                small_primes.append(m)
 
        s_m = s[m]
        threshold = limit // m
 
        for p in small_primes:
            if p > threshold:
                break
 
            if m % p:
                s[p * m] = s_m
            else:
                e, q = 2, m // p
 
                while q % p == 0:
                    e += 1
                    q //= p
 
                s[p * m] = max(s_for_prime_power(p, e), s[q])
                break
 
    for i in range(m, limit + 1):
        if s[i] == 0:
            s[i] = i
 
    return sum(s[start:])
 
 
assert div_of_factorials(0, 10) == 5
assert div_of_factorials(0, 25) == 10
assert div_of_factorials(0, 100) == 2012
 
print(div_of_factorials(6, 10))
print(div_of_factorials(6, 10000))

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