TITLE: Does the sum $\sum_{n \geq 1} \frac{2^n\operatorname{mod} n}{n^2}$ converge?
QUESTION [40 upvotes]: I am somewhat a noob, and I don't recall my math preparation from college. I know that the sum $\displaystyle \sum_{n\geq 1}\frac{1}{n}$ is divergent and my question is if the sum$$\sum \limits _{n\geq 1}\frac{2^n\mod n}{n^2}$$converges. I think is not but I do not know how to prove that! Thanks!

REPLY [5 votes]: The point of this post is to give some plots that corroborate Sangchul Lee's argument. These were produced by Mathematica
The first plot lists the numbers $(2^n\mod n)/n$ for all multiples of $5$. You see the horizontal lines at multiples of $0.2$. They were already visible in the unrestricted plot, when I didn't restrict $n$ to multiples of five, but here they are easier to spot:

When we only include the values $n=5p$, the picture is even clearer:

Multiples of five are not the only structure in there. Restricting the choice of $n$ to numbers of the form $n=7p$ gives something quite similar.