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!