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Let $n\in \Bbb N$. Any finite group $G$ with $|G|=n$ has a solution for the Diophantine equation $$\sum_{d|n}x_d\phi(d)=n~~~~~~~~~~~~~~~~~~~~~~~(1)$$ where $\phi$ is the Euler's totient function, $d$ is any positive divisor of $n$ and the $x_d$ are the unknowns (see this). Call such a solution a $G$-solution.

Are there number-theoretic conditions which characterize $G$-solutions of the equation where $G$ is a simple group?

Edit:

The comments say that I should add this:

Definition: Let $G$ be a group of order $n$. $(x_d)_{d|n}$ is a $G$-solution of equation $(1)$ if for each $d$, $x_d\phi(d)$ is the number of elements of $G$ with order $d$.

I'm looking for number-theoretic characterization of a {simple $G$}-solution, that is, a $G$-solution with $G$ a simple group.

It is clear that a $G$ produces a unique $G$-solution. I'm not completely sure, but I think a unique simple $G$ can produce a {simple $G$}-solution.

Let $n\in \Bbb N$. Any finite group $G$ with $|G|=n$ has a solution for the Diophantine equation $$\sum_{d|n}x_d\phi(d)=n~~~~~~~~~~~~~~~~~~~~~~~(1)$$ where $\phi$ is the Euler's totient function, $d$ is any positive divisor of $n$ and the $x_d$ are the unknowns (see this). Call such a solution a $G$-solution.

Are there number-theoretic conditions which characterize $G$-solutions of the equation where $G$ is a simple group?

Edit:

The comments say that I should add this:

Definition: Let $G$ be a group of order $n$. $(x_d)_{d|n}$ is a $G$-solution of equation $(1)$ if for each $d$, $x_d\phi(d)$ is the number of elements of $G$ with order $d$.

I'm looking for number-theoretic characterization of a {simple $G$}-solution, that is, a $G$-solution with $G$ a simple group.

It is clear that a $G$ produces a unique $G$-solution. I'm not completely sure, but I think a unique simple $G$ can produce a {simple $G$}-solution.

Let $n\in \Bbb N$. Any finite group $G$ with $|G|=n$ has a solution for the Diophantine equation $$\sum_{d|n}x_d\phi(d)=n~~~~~~~~~~~~~~~~~~~~~~~(1)$$ where $\phi$ is the Euler's totient function, $d$ is any positive divisor of $n$ and the $x_d$ are the unknowns (see this). Call such a solution a $G$-solution.

Are there number-theoretic conditions which characterize $G$-solutions of the equation where $G$ is a simple group?

Edit:

It seems the definition of a $G$-solution does not seem clear. So I add this:

Definition: Let $G$ be a group of order $n$. $(x_d)_{d|n}$ is a $G$-solution of equation $(1)$ if for each $d$, $x_d\phi(d)$ is the number of elements of $G$ with order $d$.

I'm looking for number-theoretic characterization of a {simple $G$}-solution, that is, a $G$-solution with $G$ a simple group.

Let $n\in \Bbb N$. Any finite group $G$ with $|G|=n$ has a solution for the Diophantine equation $$\sum_{d|n}x_d\phi(d)=n~~~~~~~~~~~~~~~~~~~~~~~(1)$$ where $\phi$ is the Euler's totient function, $d$ is any positive divisor of $n$ and the $x_d$ are the unknowns (see this). Call such a solution a $G$-solution.

Are there number-theoretic conditions which characterize $G$-solutions of the equation where $G$ is a simple group?

Edit:

The comments say that I should add this:

Definition: Let $G$ be a group of order $n$. $(x_d)_{d|n}$ is a $G$-solution of equation $(1)$ if for each $d$, $x_d\phi(d)$ is the number of elements of $G$ with order $d$.

I'm looking for number-theoretic characterization of a {simple $G$}-solution, that is, a $G$-solution with $G$ a simple group.

It is clear that a $G$ produces a unique $G$-solution. I'm not completely sure, but I think a unique simple $G$ can produce a {simple $G$}-solution.

Let $n\in \Bbb N$. Any finite group $G$ with $|G|=n$ has a solution for the Diophantine equation $$\sum_{d|n}x_d\phi(d)=n$$ where $\phi$ is the Euler's totient function, $d$ is any positive divisor of $n$ and the $x_d$ are the unknowns (see this). Call such a solution a $G$-solution.

Are there number-theoretic conditions which characterize $G$-solutions of the equation where $G$ is a simple group?

Let $n\in \Bbb N$. Any finite group $G$ with $|G|=n$ has a solution for the Diophantine equation $$\sum_{d|n}x_d\phi(d)=n~~~~~~~~~~~~~~~~~~~~~~~(1)$$ where $\phi$ is the Euler's totient function, $d$ is any positive divisor of $n$ and the $x_d$ are the unknowns (see this). Call such a solution a $G$-solution.

Are there number-theoretic conditions which characterize $G$-solutions of the equation where $G$ is a simple group?

Edit:

It seems the definition of a $G$-solution does not seem clear. So I add this:

Definition: Let $G$ be a group of order $n$. $(x_d)_{d|n}$ is a $G$-solution of equation $(1)$ if for each $d$, $x_d\phi(d)$ is the number of elements of $G$ with order $d$.

I'm looking for number-theoretic characterization of a {simple $G$}-solution, that is, a $G$-solution with $G$ a simple group.