Totient Maximum

Problem Statement

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.

Approach

The solution involves:

1. Recognizing that n/φ(n) is maximized when n has many small prime factors
2. The answer is the product of consecutive primes: 2 × 3 × 5 × 7 × 11 × 13 × 17
3. Finding the largest primorial (product of consecutive primes) ≤ 1,000,000
4. Alternatively, computing φ(n) for all n and finding the maximum ratio