project_euler.problem_072.sol1
==============================

.. py:module:: project_euler.problem_072.sol1

.. autoapi-nested-parse::

   Problem 72 Counting fractions: https://projecteuler.net/problem=72

   Description:

   Consider the fraction, n/d, where n and d are positive integers. If n<d and HCF(n,d)=1,
   it is called a reduced proper fraction.
   If we list the set of reduced proper fractions for d ≤ 8 in ascending order of size, we
   get: 1/8, 1/7, 1/6, 1/5, 1/4, 2/7, 1/3, 3/8, 2/5, 3/7, 1/2, 4/7, 3/5, 5/8, 2/3, 5/7,
   3/4, 4/5, 5/6, 6/7, 7/8
   It can be seen that there are 21 elements in this set.
   How many elements would be contained in the set of reduced proper fractions for
   d ≤ 1,000,000?

   Solution:

   Number of numbers between 1 and n that are coprime to n is given by the Euler's Totient
   function, phi(n). So, the answer is simply the sum of phi(n) for 2 <= n <= 1,000,000
   Sum of phi(d), for all d|n = n. This result can be used to find phi(n) using a sieve.

   Time: 1 sec



Functions
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.. autoapisummary::

   project_euler.problem_072.sol1.solution


Module Contents
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.. py:function:: solution(limit: int = 1000000) -> int

   Returns an integer, the solution to the problem
   >>> solution(10)
   31
   >>> solution(100)
   3043
   >>> solution(1_000)
   304191