This question was asked at MSE but never received an answer.

Let $A\subset\mathbb{N}$ be a subset of the natural numbers, and let $\sigma(n)$ denote the sum of divisors of $n$. Recall that we have the bound $\sigma(n) = O(n\log\log n)$. Consider the sums $\sum_{a\in A} 1/a$ and $\sum_{a\in A} 1/\sigma(a)$.

**Question.** Is there on $A\subset\mathbb{N}$ so that the first sum diverges but the second
sum converges ?

If the answer is already known, a reference would more than suffice. Thank you