## Is the sum $\sum_{d\mid n}\frac1{d+1}$ never integral?

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!