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About inequalities that involve the sum of divisors, the Euler's totient and the aliquot part $\sigma(n)-n$

In this post, for integers $n\geq 1$, I denote the sum of divisors $\sum_{1\leq d\mid n}d$ as $\sigma(n)$ and the Euler's totient function as $\varphi(n)$. It's easy to check* that if we assume that $...