For an integer $n>1$ in this post we denote the Dedekind psi function as $\psi(n)=n\prod_{\substack{p\mid n\\p\text{ prime}}}\left(1+\frac{1}{p}\right)$ and the product of distinct primes dividing our integer $n$ as $\operatorname{rad}(n)=\prod_{\substack{p\mid n\\p\text{ prime}}}p$, we've $\psi(1)=\operatorname{rad}(1)=1$. Both are important number theoretic functions in several subjects of mathematics. As reference I add the Wikipedia article *Dedekind psi function* (or [1]) and the corresponding Wikipedia *Radical of an integer.*

An integer $n\geq 1$ is said a perfect number if $\sigma(n)=\sum_{1\leq d\mid n}d=2n$. For example $n=6$ is the first perfect number and the only square-free (even) perfect number. It is unknown if odd perfect numbers do exist. Wikipedia has an article for *Perfect number.*

Using the properties of the aritmetic functions $\psi(n)$ and $\operatorname{rad}(n)$ one can to check easily the veracity of the following claim for even perfect numbers invoking the known as Euclid-Euler theorem.

**Claim.** *A) If* $n$ *is an even perfect number then the identity*
$$\frac{\psi(n)}{n}=\frac{3}{2}+\frac{3(1+\sqrt{1+8n})}{8n}\tag{1}$$
*holds.*

*B) If* $n$ *is an even perfect number then the identity*
$$\sum_{\substack{1\leq d\mid n\\d<\operatorname{rad}(n)}}\frac{1}{d}=2+\frac{3-\sqrt{1+8n}}{4n}\tag{2}$$
*holds.*

Previous formulas provides the best approximations of the arithmetic functions in the corresponding *LHS* of $(1)$ and $(2)$ for even perfect numbers because are identities (in terms of $n$ from the corresponding *RHS*).

Question.1) What estimation/bounds can be done in terms of $n$ about $\frac{\psi(n)}{n}$ on assumption that $n$ is an odd perfect number? 2) And, what estimation/bounds can be done in terms pf $n$ about $\sum_{\substack{1\leq d\mid n\\d<\operatorname{rad}(n)}}\frac{1}{d}$ on assumption that $n$ is an odd perfect number?Many thanks.

I don't know if the problem to find bounds for the quantity $\prod_{\substack{p\mid n\\p\text{ prime}}}\left(1+\frac{1}{p}\right)$, when one assumes that $n$ is an odd perfect number, is in the literature, in this case feel free to refer the literature answering the **Question 1)** as a reference request and I try to find and read those statements from the literature.

As motivation for those expressions I refer that the expression $(1)$ arises just from the quotient of the specialization of $\psi(n)$ for even perfect numbers by $n$, and on the other hand the sum in the *LHS* of $(2)$ evokes in some manner a variant of the sum $\sum_{1\leq d\mid n}\frac{1}{d}$ for perfect numbers or other more specific sums that are in the literature of odd perfect numbers.

## References:

[1] Tom M. Apostol, *Introduction to Analytic Number Theory*, Undergraduate Texts in Mathematics Springer (1976).

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