Problem 69

Totient 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.

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These problems are part of
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