Recall that a positive integer $n$ is a perfect number if and only if $$\frac{\sigma(n)}n=\sum_{d\mid n}\frac1d=2.$$

QUESTION: Is my following conjecture true?

**Conjecture**. (i) We have $\sum_{d\mid n}\frac1{d+1}\not\in\mathbb Z$ for all $n=1,2,3,\ldots$. Moreover, for any positive integers $k$ and $m$, all the numbers
$$\sum_{d\mid n}\frac1{(d+m)^k}\ \ (n=1,2,3,\ldots)$$
have pairwise distinct fractional parts, and none of them is an integer.

(ii) For any integer $k>1$, all the numbers $$\sum_{d\mid n}\frac1{d^k}\ \ (n=1,2,3,\ldots)$$ have pairwise distinct fractional parts.

I formulated this conjecture in October 2015 on the basis of my computation. Your comments are welcome!

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