mirror of
https://github.com/TheAlgorithms/Python.git
synced 2024-11-24 21:41:08 +00:00
24d3cf8244
* The black formatter is no longer beta
* pre-commit autoupdate
* pre-commit autoupdate
* Remove project_euler/problem_145 which is killing our CI tests
* updating DIRECTORY.md
Co-authored-by: github-actions <${GITHUB_ACTOR}@users.noreply.github.com>
67 lines
1.7 KiB
Python
67 lines
1.7 KiB
Python
"""
|
|
Totient maximum
|
|
Problem 69: https://projecteuler.net/problem=69
|
|
|
|
Euler's Totient function, φ(n) [sometimes called the phi function],
|
|
is used to determine the number of numbers less than n which are relatively prime to n.
|
|
For example, as 1, 2, 4, 5, 7, and 8,
|
|
are all less than nine and relatively prime to nine, φ(9)=6.
|
|
|
|
n Relatively Prime φ(n) n/φ(n)
|
|
2 1 1 2
|
|
3 1,2 2 1.5
|
|
4 1,3 2 2
|
|
5 1,2,3,4 4 1.25
|
|
6 1,5 2 3
|
|
7 1,2,3,4,5,6 6 1.1666...
|
|
8 1,3,5,7 4 2
|
|
9 1,2,4,5,7,8 6 1.5
|
|
10 1,3,7,9 4 2.5
|
|
|
|
It can be seen that n=6 produces a maximum n/φ(n) for n ≤ 10.
|
|
|
|
Find the value of n ≤ 1,000,000 for which n/φ(n) is a maximum.
|
|
"""
|
|
|
|
|
|
def solution(n: int = 10**6) -> int:
|
|
"""
|
|
Returns solution to problem.
|
|
Algorithm:
|
|
1. Precompute φ(k) for all natural k, k <= n using product formula (wikilink below)
|
|
https://en.wikipedia.org/wiki/Euler%27s_totient_function#Euler's_product_formula
|
|
|
|
2. Find k/φ(k) for all k ≤ n and return the k that attains maximum
|
|
|
|
>>> solution(10)
|
|
6
|
|
|
|
>>> solution(100)
|
|
30
|
|
|
|
>>> solution(9973)
|
|
2310
|
|
|
|
"""
|
|
|
|
if n <= 0:
|
|
raise ValueError("Please enter an integer greater than 0")
|
|
|
|
phi = list(range(n + 1))
|
|
for number in range(2, n + 1):
|
|
if phi[number] == number:
|
|
phi[number] -= 1
|
|
for multiple in range(number * 2, n + 1, number):
|
|
phi[multiple] = (phi[multiple] // number) * (number - 1)
|
|
|
|
answer = 1
|
|
for number in range(1, n + 1):
|
|
if (answer / phi[answer]) < (number / phi[number]):
|
|
answer = number
|
|
|
|
return answer
|
|
|
|
|
|
if __name__ == "__main__":
|
|
print(solution())
|