Skip to main content
Proofreading; `|` -> `\mid`
Source Link
LSpice
  • 12.9k
  • 4
  • 45
  • 69

Are there infiniteinfinitely many positive integers $\ n\ $ satisfying $\ \varphi(n)|\sigma\mid \sigma(n)\ $$?

Are there infinitely many postive integers $\ n\ $ satisfying $$\varphi(n)|\sigma(n)$$$$\varphi(n)\mid \sigma(n)$$ where $\varphi(n)$ is Euler’s totient function,and and $\sigma(n)$ is the sum of divisors of $n$  ? If yes  , can it be proven  ?

The first few such $\ n\ $ are $$n=1, 2, 3, 6, 12, 14, 15, 30, 35, 42, 56, 70, 78, 105, 140, 168, 190$$$$n=1, 2, 3, 6, 12, 14, 15, 30, 35, 42, 56, 70, 78, 105, 140, 168, 190.$$

Ideas for a proof of the existence of infiniteinfinitely many such positive integers (like families of such numbers probably being infinite) are appreciated.

I posted this question to mathstackMath.StackExchange, but I couldn't get reaction. That is why I'm asking this here. If this is not suitable for this place, please answer in mathstack format Math.StackExchange: https://math.stackexchange.com/questions/4407570/are-there-infinite-many-positive-integers-n-satisfying-varphin-sigm?noredirect=1&lq=1

Thank you in advance.Are there infinite many positive integers $\ n\ $ satisfying $\ \varphi(n)|\sigma(n)\ $?

Are there infinite many positive integers $\ n\ $ satisfying $\ \varphi(n)|\sigma(n)\ $?

Are there infinitely many postive integers $\ n\ $ satisfying $$\varphi(n)|\sigma(n)$$ where $\varphi(n)$ is Euler’s totient function,and $\sigma(n)$ is the sum of divisors of $n$  ? If yes  , can it be proven  ?

The first few such $\ n\ $ are $$n=1, 2, 3, 6, 12, 14, 15, 30, 35, 42, 56, 70, 78, 105, 140, 168, 190$$

Ideas for a proof of the existence of infinite many such positive integers (like families of such numbers probably being infinite) are appreciated.

I posted this question to mathstack, but I couldn't get reaction. That is why I'm asking this here. If this is not suitable for this place, please answer in mathstack form. https://math.stackexchange.com/questions/4407570/are-there-infinite-many-positive-integers-n-satisfying-varphin-sigm?noredirect=1&lq=1

Thank you in advance.

Are there infinitely many positive integers $\ n\ $ satisfying $\ \varphi(n)\mid \sigma(n)$?

Are there infinitely many postive integers $\ n\ $ satisfying $$\varphi(n)\mid \sigma(n)$$ where $\varphi(n)$ is Euler’s totient function, and $\sigma(n)$ is the sum of divisors of $n$? If yes, can it be proven?

The first few such $\ n\ $ are $$n=1, 2, 3, 6, 12, 14, 15, 30, 35, 42, 56, 70, 78, 105, 140, 168, 190.$$

Ideas for a proof of the existence of infinitely many such positive integers (like families of such numbers probably being infinite) are appreciated.

I posted this question to Math.StackExchange, but I couldn't get reaction. That is why I'm asking this here. If this is not suitable for this place, please answer at Math.StackExchange: Are there infinite many positive integers $\ n\ $ satisfying $\ \varphi(n)|\sigma(n)\ $?

Source Link
math110
  • 4.3k
  • 18
  • 46

Are there infinite many positive integers $\ n\ $ satisfying $\ \varphi(n)|\sigma(n)\ $?

Are there infinitely many postive integers $\ n\ $ satisfying $$\varphi(n)|\sigma(n)$$ where $\varphi(n)$ is Euler’s totient function,and $\sigma(n)$ is the sum of divisors of $n$ ? If yes , can it be proven ?

The first few such $\ n\ $ are $$n=1, 2, 3, 6, 12, 14, 15, 30, 35, 42, 56, 70, 78, 105, 140, 168, 190$$

Ideas for a proof of the existence of infinite many such positive integers (like families of such numbers probably being infinite) are appreciated.

I posted this question to mathstack, but I couldn't get reaction. That is why I'm asking this here. If this is not suitable for this place, please answer in mathstack form. https://math.stackexchange.com/questions/4407570/are-there-infinite-many-positive-integers-n-satisfying-varphin-sigm?noredirect=1&lq=1

Thank you in advance.