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Problem 69


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.


欧拉总计函数与最大值

在小于n的数中,与n互质的数的数目记为欧拉总计函数φ(n)(有时也称为φ函数)。例如,因为1、2、4、5、7和8均小于9且与9互质,故φ(9)=6。

n 互质的数 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

可以看出,对于n ≤ 10,当n=6时n/φ(n)取得最大值。

当n ≤ 1,000,000时,求使得n/φ(n)取得最大值的n。