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This afternoon I am doing experiments. Then I consider the following arithmetic functions

$$\sigma_{+}(n)=\sum_{\substack{k\mid n \\ k\geq \sqrt{n}}}k,$$ the sum of those divisors $k$ grater or equal than the square root of an integer $n$, and $$\phi_{-}(n)=\sum_{\substack{(k,n)=1\\ k<\sqrt{n}}}1$$ thus is the number of coprime integers $1\leq k$ that are less than (strictly) the square root of an integer $n$. I don't know if the following questions are easys or well known:

Question 1. After a lot of experiments and inspired in an statement corcerning the sum of divisor function and Euler's totient function, I asked myself if it is possible determine the superior limit (I believe that as $1$) or the inferior limit (then I believe that is $6/\pi^2$) of $$\frac{\sigma_{+}(n)\phi_{-}(n)}{n^{3/2}}.$$ I don't know if the question is easy, if it is obvious you can state only or well the inferior limit or the superior limit using the well known inequalities from the literature, involving the sum of divisors functions and the Euler's totient function.

After I saw that it is easy to chek that if $n$ is prime then $$\sigma_{+}(n) \equiv 0 \text{ mod }\phi_{-}(n), $$ I've asked if it is possible solve the following

Question 2. Can you show that $n$ is composite and satistifies $\sigma_{+}(n) \equiv 0 \text{ mod }\phi_{-}(n)$ if and only if $n\in\text{A182147}$ : Numbers n equal to the sum of its proper divisors greater than the square root of n?

Previous reference A182147 is for the sequence that starts as 42, 54, 66, 78, 102, 114, 138..., you can see in The On-Line Encyclopedia of Integer Sequences. Many thanks.

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  • $\begingroup$ Very thanks much for the vote up, because I don't know if it is a right questions, truly I don't know if are obvious. $\endgroup$ – user243301 Sep 1 '16 at 16:49

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