p069.py

# -*- coding: utf-8 -*-
"""
Created on Tue Jul 14 15:50:43 2020

@author: zhixia liu
"""

"""
Project Euler 69: Toient Maximum

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.

"""

#%% naive bf

from helper import totient

maxratio = 1

for i in range(2,1000001):
    r = i/totient(i)
    if r>maxratio:
        maxratio=r
        print(i,r)
        
#%% by hand

n = 2*3*5*7*11*13*17