Is the asymptotic density of positive integers $n$ satisfying $\gcd(n, \sigma(n^2))=\gcd(n^2, \sigma(n^2))$ equal to zero?

Question

(This post is an offshoot of this MSE question.)

Let $\sigma(x)$ denote the sum of divisors of $x$. (https://oeis.org/A000203)

QUESTION

Is the asymptotic density of positive integers $n$ satisfying $\gcd(n, \sigma(n^2))=\gcd(n^2, \sigma(n^2))$ equal to zero?

I tried searching for examples and counterexamples to the equation $$\gcd(n, \sigma(n^2))=\gcd(n^2, \sigma(n^2))$$ via Sage Cell Server, it gave me this output for the following Pari-GP script:

for(x=1, 100, if(gcd(x,sigma(x^2))==gcd(x^2,sigma(x^2)),print(x)))

All positive integers from $1$ to $100$ (except for the integer $99$) satisfy $\gcd(n, \sigma(n^2))=\gcd(n^2, \sigma(n^2))$.

Generalizing the first (counter)example of $99$ is trivial.

If ${3^2}\cdot{11} \parallel n$, then $11 \parallel \gcd(n,\sigma(n^2))$ and $11^2 \parallel \gcd(n^2,\sigma(n^2))$. So the asymptotic density in question is less than $$1-\frac{2}{3^3}\cdot\frac{10}{11^2} = \frac{3247}{3267} \approx 0.993878.$$

Also, if $3 \parallel n$, then with probability $1$ there exist two distinct primes $y$ and $z$ congruent to $1$ modulo $3$ such that $y \parallel n$ and $z \parallel n$. In this case, we get $3 \parallel \gcd(n,\sigma(n^2))$ and $3^2 \parallel \gcd(n^2,\sigma(n^2))$. So the asymptotic density in question is less than $$1-\frac{2}{3^2} = \frac{7}{9} \approx 0.\overline{777}.$$

The real open problem is whether the asymptotic density is $0$.

Answer 1

I think the density does go to zero, but quite slowly. If $p \equiv 1 \bmod 6$ is prime then there are two solutions $0<r<s<p-1$ of $$x^2+x+1=0 \bmod p$$

If $p\parallel n$ then, with probability $1,$ there are two distinct primes $x $ and $ y,$ each congruent to $r \bmod p,$ with $x \parallel n$ and $y \parallel n.$ ( Either or both could be congruent to $s$ as well.)

Then $p \parallel \gcd(n,\sigma(n^2))$ while $p^2 \parallel \gcd(n^2,\sigma(n^2)).$ So the asymptotic density for this not to happen is $1-\frac{p-1}{p^2}<1-\frac{1}{p+2}$

If we can argue that the chance that none of these events happen is asymptotically $\prod(1-\frac{p-1}{p^2})$ over the primes congruent to $1 \bmod 6,$ then that asymptotic density is $0.$