Problem 245

Problem 245


We shall call a fraction that cannot be cancelled down a resilient fraction. Furthermore we shall define the resilience of a denominator, R(d), to be the ratio of its proper fractions that are resilient; for example, R(12) = 411.

The resilience of a number d > 1 is then$\frac{\phi(d)}{d-1}$ where φ is Euler’s totient function.

We further define the coresilience of a number n > 1 as C(n)=$\frac{n − \phi(n)}{n-1}$.

The coresilience of a prime p is C(p)=$\frac{1}{p-1}$.

Find the sum of all composite integers 1 < n ≤ 2×1011, for which C(n) is a unit fraction.


我们称不能被约简的分数为不可约分数。进一步地,我们可以定义分母的不可约度R(d)为它的真分数中不可约分数的比例;例如,R(12) = 4/11

数d > 1的不可约度可以表示成$\frac{\phi(d)}{d-1}$,其中φ是欧拉总计函数。

我们进一步定义数n > 1的可约度为C(n)=$\frac{n − \phi(n)}{n-1}$。


在所有1 < n ≤ 2×1011合数中,有些数满足C(n)约简后为单位分数,求这些数之和。