Expert · Number theory · 237 of 264
Counting primes fast
Count the prime numbers up to n, for n as large as 2,000,000. Checking each number separately is too slow: use the sieve of Eratosthenes.
Input: n. Output: the number of primes up to n.
Examples
Input
10
Output
4
Input
1
Output
0
Input
100
Output
25
One more test checks other inputs when you press Check.
Stuck? The idea behind number theory
Whole-number tricks: n % d == 0 means d divides n; test divisors only up to the square root; the sieve of Eratosthenes finds every prime up to n at once.
pow(a, b, m) works out a to the power b, modulo m, quickly, even when the power is huge.
n = 50
is_prime = [True] * (n + 1)
is_prime[0] = is_prime[1] = False
for i in range(2, int(n ** 0.5) + 1):
if is_prime[i]:
for j in range(i * i, n + 1, i):
is_prime[j] = False
print(sum(is_prime))Watch for: Testing every number up to n one by one is too slow for big inputs: the hidden tests notice.