sequences and series - Convergent sum of reciprocals?

Let n denote a positive integer and let $\sigma(n)$ denote the sum
of all divisors of $n$, so that $\sigma(n)$ is larger than $n$ (for $n > 1$)
but not by much since it's bounded above by $c\ n\log \log n$ .

This invites comparison between situations in which the two are
interchanged. As an example, consider the following:

Let $A$ be a subset of the positive integers. Suppose that
the sum of $1/n$ , for all $n$ in $A$ , converges. Then the sum

of 1/$\sigma(n)$ , over all $n$ in $A$, clearly converges as well.

Is the converse true?