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Problem 70


Problem 70


Totient permutation

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.

The number 1 is considered to be relatively prime to every positive number, so φ(1)=1.

Interestingly, φ(87109)=79180, and it can be seen that 87109 is a permutation of 79180.

Find the value of n, 1 < n < 107, for which φ(n) is a permutation of n and the ratio n/φ(n) produces a minimum.


欧拉总计函数与重排

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。

1被认为和任意正整数互质,所以φ(1)=1。

有趣的是,φ(87109)=79180,而79180恰好是87109的一个重排。

在1 < n < 107中,有些n满足φ(n)是n的一个重排,求这些取值中使n/φ(n)最小的一个。