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Let $\varphi$ be Euler's totient function, and let $\sigma(n)=\sum_{d\mid n}d$ for $n=1,2,3,\ldots$. Both $\varphi$ and $\sigma$ are multiplicative functions. It is easy to see that the numbers $\varphi(n^2)=n\varphi(n)\ (n=1,2,3,\ldots)$ are pairwise distinct.

Question. Are the numbers $\varphi(n^2)\sigma(n^2)\ (n=1,2,3,\ldots)$ pairwise distinct? Are the rationals $\sigma(n^2)/\varphi(n^2)\ (n=1,2,3,\ldots)$ pairwise distinct?

I had this question in 2015, and conjecture that the answer is affirmative. I have verified that $$\varphi(n^2)\sigma(n^2)\ \ (n=1,\ldots,10^7)$$ are indeed pairwise distinct, and also $$\frac{\sigma(n^2)}{\varphi(n^2)}\ \ (n=1,\ldots,10^6)$$ are pairwise distinct.

Any ideas to solve the problem? Your comments are welcome!

Let $\varphi$ be Euler's totient function, and let $\sigma(n)=\sum_{d\mid n}d$ for $n=1,2,3,\ldots$. Both $\varphi$ and $\sigma$ are multiplicative functions. It is easy to see that the numbers $$\varphi(n^2)=n\varphi(n)\ \ (n=1,2,3,\ldots)$$ are pairwise distinct.

Question 1. Are the numbers $\varphi(n^2)\sigma(n^2)\ (n=1,2,3,\ldots)$ pairwise distinct?

Question 2. Are the rationals $\sigma(n^2)/\varphi(n^2)\ (n=1,2,3,\ldots)$ pairwise distinct?

I had the questions in 2015, and conjecture that the answers to both questions are affirmative. I have verified that $$\varphi(n^2)\sigma(n^2)\ \ (n=1,\ldots,10^7)$$ are indeed pairwise distinct, and also $$\frac{\sigma(n^2)}{\varphi(n^2)}\ \ (n=1,\ldots,10^7)$$ are pairwise distinct.

Any ideas to solve the problem? Your comments are welcome!

Let $\varphi$ be Euler's totient function, and let $\sigma(n)=\sum_{d\mid n}d$ for $n=1,2,3,\ldots$. Both $\varphi$ and $\sigma$ are multiplicative functions. It is easy to see that the numbers $\varphi(n^2)=n\varphi(n)\ (n=1,2,3,\ldots)$ are pairwise distinct.

Question. Are the numbers $\varphi(n^2)\sigma(n^2)\ (n=1,2,3,\ldots)$ pairwise distinct? Are the rationals $\sigma(n^2)/\varphi(n^2)\ (n=1,2,3,\ldots)$ pairwise distinct?

I had this question in 2015, and conjecture that the answer is affirmative. I have verified that $$\varphi(n^2)\sigma(n^2)\ \ (n=1,\ldots,10^6)$$ are indeed pairwise distinct, and also $$\frac{\sigma(n^2)}{\varphi(n^2)}\ \ (n=1,\ldots,10^6)$$ are pairwise distinct.

Any ideas to solve the problem? Your comments are welcome!

Let $\varphi$ be Euler's totient function, and let $\sigma(n)=\sum_{d\mid n}d$ for $n=1,2,3,\ldots$. Both $\varphi$ and $\sigma$ are multiplicative functions. It is easy to see that the numbers $\varphi(n^2)=n\varphi(n)\ (n=1,2,3,\ldots)$ are pairwise distinct.

Question. Are the numbers $\varphi(n^2)\sigma(n^2)\ (n=1,2,3,\ldots)$ pairwise distinct? Are the rationals $\sigma(n^2)/\varphi(n^2)\ (n=1,2,3,\ldots)$ pairwise distinct?

I had this question in 2015, and conjecture that the answer is affirmative. I have verified that $$\varphi(n^2)\sigma(n^2)\ \ (n=1,\ldots,10^7)$$ are indeed pairwise distinct, and also $$\frac{\sigma(n^2)}{\varphi(n^2)}\ \ (n=1,\ldots,10^6)$$ are pairwise distinct.

Any ideas to solve the problem? Your comments are welcome!

Let $\varphi$ be Euler's totient function, and let $\sigma(n)=\sum_{d\mid n}d$ for $n=1,2,3,\ldots$. Both $\varphi$ and $\sigma$ are multiplicative functions. It is easy to see that the numbers $\varphi(n^2)=n\varphi(n)\ (n=1,2,3,\ldots)$ are pairwise distinct.

Question. Are the numbers $\varphi(n^2)\sigma(n^2)\ (n=1,2,3,\ldots)$ pairwise distinct? Are the rationals $\sigma(n^2)/\varphi(n^2)\ (n=1,2,3,\ldots)$ pairwise distinct?

I had this question in 2015, and conjecture that the answer is affirmative. I have verified that $$\varphi(n^2)\sigma(n^2)\ \ (n=1,\ldots,80000)$$ are indeed pairwise distinct, and also $$\frac{\sigma(n^2)}{\varphi(n^2)}\ \ (n=1,\ldots,80000)$$ are pairwise distinct.

Any ideas to solve the problem? Your comments are welcome!

Let $\varphi$ be Euler's totient function, and let $\sigma(n)=\sum_{d\mid n}d$ for $n=1,2,3,\ldots$. Both $\varphi$ and $\sigma$ are multiplicative functions. It is easy to see that the numbers $\varphi(n^2)=n\varphi(n)\ (n=1,2,3,\ldots)$ are pairwise distinct.

Question. Are the numbers $\varphi(n^2)\sigma(n^2)\ (n=1,2,3,\ldots)$ pairwise distinct? Are the rationals $\sigma(n^2)/\varphi(n^2)\ (n=1,2,3,\ldots)$ pairwise distinct?

I had this question in 2015, and conjecture that the answer is affirmative. I have verified that $$\varphi(n^2)\sigma(n^2)\ \ (n=1,\ldots,10^6)$$ are indeed pairwise distinct, and also $$\frac{\sigma(n^2)}{\varphi(n^2)}\ \ (n=1,\ldots,10^6)$$ are pairwise distinct.

Any ideas to solve the problem? Your comments are welcome!