Find all prime numbers up to a given number
Problem Statement
How do we find all the prime numbers up to a given number?
For example, when n = 10, the output should be 4 since there are 4 prime numbers less than 10, namely 2, 3, 5, 7.
Naive Approach
A naive approach would be to iterate from 2 to n and count the number of primes along the way:
python3 def count_primes_less_than(n: int) -> int: return sum(is_prime(i) for i in range(2, n))
def is_prime(num: int) -> bool: return all(num % i != 0 for i in range(2, num))
We can improve the running time of is_prime from O(n) to O(sqrt(n)):
python3 from math import sqrt
def is_prime(num: int) -> bool: return all(num % i != 0 for i in range(2, int(sqrt(num)) + 1))
Sieve of Eratosthenes
An efficient approach is called Sieve of Eratosthenes. Below is a Python implementation:
python3 from math import sqrt
def count_primes_less_than(n: int) -> int: nums = [True] * n ans = 0 for i in range(2, n): if nums[i]: ans += 1 nums[i] = False for j in range(i * i, n, i): nums[j] = False return ans
Another implementation:
python3 from math import sqrt
def count_primes_less_than(n: int) -> int: if n < 2: return 0 composites = set() for i in range(2, int(sqrt(n)) + 1): if i not in composites: for j in range(i * i, n, i): composites.add(j) return n - len(composites) - 2
Prime Arrangements
Now that we know how to efficiently find the number of primes less than a given integer, let’s count the number of permutations of 1 to n so that prime numbers are at prime indices (1-indexed)].
For example, when n=9, we want primes <= 9 to be at indices 2, 3, 5, 7 and composites at indices 1, 4, 6, 8, 9 (1-indexed). We can place primes <= 9 at indices 2, 3, 5, 7 in 4! = 24 ways. We can place composites <= 9 at indices 1, 4, 6, 8, 9 in 5! = 120 ways.
In general, the prime arrangement count equals the number of permutations of the primes multipled by the number of permutations of the composites:
python3 from math import factorial
def numPrimeArrangements(n: int) -> int: primes = count_primes_less_than(n + 1) composites = n - primes return factorial(primes) * factorial(composites)