project_euler.problem_069.sol1

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.

Functions

solution(→ int)

Returns solution to problem.

Module Contents

project_euler.problem_069.sol1.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

  1. Find k/φ(k) for all k ≤ n and return the k that attains maximum

>>> solution(10)
6

>>> solution(100)
30

>>> solution(9973)
2310