If $a_n$ is a sequence of non-increasing positive numbers, then suppose we already know that $$\sum_p a_p$$ converges, when $p$ runs over the primes, what should be used to prove that $$\sum_n \frac{a_n}{\log{n}}$$ also converges, where $n$ runs over the positive naturals?
And also, how to show the converse is also true?