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Let $\varphi(m)$ the Euler's totient function and $\sigma(m)$ the sum of divisors function.

If $n$ is an odd perfect number then $n$ satisfies $$\varphi(n)=\varphi(\sigma(n)).\tag{1}$$

The sequence of integers satisfying $(1)$ is defined in The On-Line Encyclopedia of Integer Sequences as sequence A006872 (see also the references).

Question. I would like to know if in the context of our condition $(1)$ we can set a lower bound and an upper bound $$\text{Lower bound}<\pi(2n)-\pi(n)<\text{Upper bound}\tag{2}$$ where $\pi(x)$ denotes the prime-counting function, and $n$ being an odd perfect number. Many thanks.

I hope that the identity $(1)$ is a good motivation for asking my Question.

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