Project Euler

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# [[Problem 69]]

## Totient Maximum

Euler's totient function, $\phi(n)$ [sometimes called the phi function], is defined as the number of positive integers not exceeding $n$ which are relatively prime to $n$. For example, as $1$, $2$, $4$, $5$, $7$, and $8$, are all less than or equal to nine and relatively prime to nine, $\phi(9)=6$.

| **$n$** | **Relatively Prime** | **$\phi(n)$** | **$n/\phi(n)$** |

|--|--|--|--|

| 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 $n/\phi(n)$ for $n\leq 10$.

Find the value of $n\leq 1\,000\,000$ for which $n/\phi(n)$ is a maximum.

## トーティエント関数の最大値

オイラーのトーティエント関数, φ(n) [時々ファイ関数とも呼ばれる]は, n と互いに素な n 未満の数の数を定める. たとえば, 1, 2, 4, 5, 7, そして8はみな9未満で9と互いに素であり, φ(9)=6.

| n | 互いに素な数 | 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 |

n ≤ 10 では n/φ(n) の最大値は n=6 であることがわかる.

n ≤ 1,000,000で n/φ(n) が最大となる値を見つけよ.

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