Coresilience

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, $R(12) = \dfrac{4}{11}$.
The resilience of a number $d \gt 1$ is then $\dfrac{\varphi(d)}{d – 1}$, where $\varphi$ is Euler’s totient function.
We further define the coresilience of a number $n \gt 1$ as $C(n) = \dfrac{n – \varphi(n)}{n – 1}$.
The coresilience of a prime $p$ is $C(p) = \dfrac{1}{p – 1}$.
Find the sum of all composite integers $1 \lt n \le 2 \times 10^{11}$, for which $C(n)$ is a unit fractionA fraction with numerator $1$.

This problem is based on advanced number theory concepts.

1. Euler’s Totient Function ($\varphi$ function): It counts the positive integers up to a given integer $n$ that are relatively prime to $n$. For example, if $n=9$, there are 6 numbers below 9 which are relatively prime to 9 (1, 2, 4, 5, 7, 8), so $\varphi(9)=6$.

2. Unit Fraction: A fraction with numerator 1.

In the problem, it’s given that the coresilience of a prime $p$ is $C(p)=\frac{1}{p-1}$, thus a unit fraction.

Now, we want to find the sum of all composites $n$ such that $C(n)$ is a unit fraction. We already have the relationship $C(n) = \frac{n – \varphi(n)}{n – 1}$.

If $C(n)$ is a unit fraction, i.e $C(n)=\frac{1}{m}$
, we can have
$\frac{n – \varphi(n)}{n – 1} = \frac{1}{m}$,
which simplifies to
$n = m \cdot \varphi(n) + m – 1$,

This equation tells that $n$ is a multiple of $m$, and $m-1$ goes into $n$ as well.

To find $n$, we have to come up with solutions for $m$ that satisfy the above two conditions. We can do this by trial and error, by using a computer program up to the upper limit 2e11 as per the question.

Note: While this will solve the problem, it is computationally expensive, especially since we’re considering composites up to $2 \times 10^{11}$, and this becomes a programming problem involving range-based searching, prime factorizations, and Euler’s Totient function.

It is important to understand that such calculations might not be feasible by hand, and hence they are usually done through computer programs or software.

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