TITLE: What is the sum of the reciprocal of all of the factors of a number?
QUESTION [12 upvotes]: Suppose I have some operation $f(n)$ that is given as

$$f(n)=\sum_{k\ge1}\frac1{a_k}$$

Where $a_k$ is the $k$th factor of $n$.
For example, $f(100)=\frac11+\frac12+\frac14+\frac15+\frac1{10}+\frac1{20}+\frac1{25}+\frac1{50}+\frac1{100}=\frac{217}{100}$
$f(101)=\frac11+\frac1{101}=\frac{102}{101}$
$f(102)=\frac11+\frac12+\frac13+\frac16+\frac1{17}+\frac1{34}+\frac1{51}+\frac1{102}=\frac{216}{102}$
I was wondering if it were possible to plot a graph of $f(n)$ and wondered if there were any interesting patterns.  I was also wondering if there is a closed form representation and if $\lim_{n\to\infty}f(n)$ could be evaluated or determined to be finite or not or any other interesting things that might happen in this limit.
Secondly, I was wondering about another similar series, which considers $b_k$ as the $k$th prime factor of $n$.

$$p(n)=\sum_{k\ge1}\frac1{b_k}$$

What can we determine about this series?

REPLY [3 votes]: Here is the plot of $f(n)=\frac{\sigma(n)}{n}$ .
Patterns in these plots are amazing!
For $1000$  
 
And here is for $100,000$

This diagram shows the $\lim_{x\to \infty} f_n$ isn't plausible !