Python/project_euler/problem_234/sol1.py
Christian Clauss 24d3cf8244
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Python

"""
https://projecteuler.net/problem=234
For an integer n ≥ 4, we define the lower prime square root of n, denoted by
lps(n), as the largest prime ≤ √n and the upper prime square root of n, ups(n),
as the smallest prime ≥ √n.
So, for example, lps(4) = 2 = ups(4), lps(1000) = 31, ups(1000) = 37. Let us
call an integer n ≥ 4 semidivisible, if one of lps(n) and ups(n) divides n,
but not both.
The sum of the semidivisible numbers not exceeding 15 is 30, the numbers are 8,
10 and 12. 15 is not semidivisible because it is a multiple of both lps(15) = 3
and ups(15) = 5. As a further example, the sum of the 92 semidivisible numbers
up to 1000 is 34825.
What is the sum of all semidivisible numbers not exceeding 999966663333 ?
"""
import math
def prime_sieve(n: int) -> list:
"""
Sieve of Erotosthenes
Function to return all the prime numbers up to a certain number
https://en.wikipedia.org/wiki/Sieve_of_Eratosthenes
>>> prime_sieve(3)
[2]
>>> prime_sieve(50)
[2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47]
"""
is_prime = [True] * n
is_prime[0] = False
is_prime[1] = False
is_prime[2] = True
for i in range(3, int(n**0.5 + 1), 2):
index = i * 2
while index < n:
is_prime[index] = False
index = index + i
primes = [2]
for i in range(3, n, 2):
if is_prime[i]:
primes.append(i)
return primes
def solution(limit: int = 999_966_663_333) -> int:
"""
Computes the solution to the problem up to the specified limit
>>> solution(1000)
34825
>>> solution(10_000)
1134942
>>> solution(100_000)
36393008
"""
primes_upper_bound = math.floor(math.sqrt(limit)) + 100
primes = prime_sieve(primes_upper_bound)
matches_sum = 0
prime_index = 0
last_prime = primes[prime_index]
while (last_prime**2) <= limit:
next_prime = primes[prime_index + 1]
lower_bound = last_prime**2
upper_bound = next_prime**2
# Get numbers divisible by lps(current)
current = lower_bound + last_prime
while upper_bound > current <= limit:
matches_sum += current
current += last_prime
# Reset the upper_bound
while (upper_bound - next_prime) > limit:
upper_bound -= next_prime
# Add the numbers divisible by ups(current)
current = upper_bound - next_prime
while current > lower_bound:
matches_sum += current
current -= next_prime
# Remove the numbers divisible by both ups and lps
current = 0
while upper_bound > current <= limit:
if current <= lower_bound:
# Increment the current number
current += last_prime * next_prime
continue
if current > limit:
break
# Remove twice since it was added by both ups and lps
matches_sum -= current * 2
# Increment the current number
current += last_prime * next_prime
# Setup for next pair
last_prime = next_prime
prime_index += 1
return matches_sum
if __name__ == "__main__":
print(solution())