# Estimation of the number of solutions for the equation $\sigma(\varphi(n))=\sigma(\operatorname{rad}(n))$

For integers $$n\geq 1$$ in this post we denote the square-free kernel as $$\operatorname{rad}(n)=\prod_{\substack{p\mid n\\p\text{ prime}}}p,$$ that is the product of distinct primes dividing an integer $$n>1$$ with the definition $$\operatorname{rad}(1)=1$$ (the Wikipedia's article dedicated to this multiplicative function is Radical of an integer).

And I denote the Euler's totient function as $$\varphi(n)$$. We consider the solutions over integers $$n\geq 1$$ of the equation $$\sigma(\varphi(n))=\sigma(\operatorname{rad}(n)).\tag{1}$$

The first few solutions are $$n=1,4,18,87,260,362$$ and $$732$$. We denote the set of all solutions of $$(1)$$ as $$\mathcal{A}$$ and for a real number $$x>1$$ let $$\mathcal{A}(x)$$ defined as $$\mathcal{A}(x)=\mathcal{A}\cap[1,x]$$ with cardinality denoted as $$\#\mathcal{A}(x)$$.

After I did a table using a program with my computer and since my equation is a variant of an equation from the literature [1], I wrote next conjecture (any case I believe that it can be wrong thus I am asking my Question).

Conjecture. The estimate $$\#\mathcal{A}(x)=O\left(\frac{x}{(\log x)^3}\right)\tag{2}$$ is true for enough large $$x>1$$.

Question. Can you prove or refute previous Conjecture? Many thanks.

I'm curious to know if we can refute previous conjeture, what methods/reasonings can one use to refute it?

## References:

[1] Jean-Marie De Koninck and Florian Luca, Positive Integers $$n$$ Such That $$\sigma(\phi(n))=\sigma(n)$$, Journal of Integer Sequences, Vol. 11 (2008), Article 08.1.5.

• Upto $10^8$ , $2\ 942$ solutions exist – Peter Jun 7 '18 at 11:17
• Many thanks for your calculation @Peter – user243301 Jun 7 '18 at 14:28
• My suggestion is that you walk through the method in the paper you cited, and try on your equation. – Sungjin Kim Jun 24 '18 at 13:34
• Thanks for your suggestion @i707107 – user243301 Jun 24 '18 at 18:09