#!/usr/bin/python3

# 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?

from timeit import default_timer
from projecteuler import sieve, phi

def main():
    start = default_timer()

    N = 1000001

    count = 0
    primes = sieve(N)

#   For any denominator d, the number of reduced proper fractions is 
#   the number of fractions n/d where gcd(n, d)=1, which is the definition
#   of Euler's Totient Function phi. It's sufficient to calculate phi for each
#   denominator and sum the value.
    for i in range(2, N):
        count = count + phi(i, primes)

    end = default_timer()

    print('Project Euler, Problem 72')
    print('Answer: {}'.format(count))

    print('Elapsed time: {:.9f} seconds'.format(end - start))

if __name__ == '__main__':
    main()