Sum of Sums over Primes that Divide the Index

[drewfstr314]

I have seen double sums, but have come across a problem involving sums over primes. However, this sum is inside a second sum, and is taken over all primes that divide the outside index, like this:

$\sum_{k=1}^{n} \sum_{p | k} \frac 1p$

for p prime.

Is there any way to manipulate this? Any help would be appreciated.

Thanks!

 

[acabus]

I think that's equivalent to $\sum_{p=2}^{n} \frac{\left \lfloor n/p \right \rfloor}{p}$, where the square brackets represent the floor function, and p runs through the primes less than or equal to n.

I don't know if that helps at all, and no doubt it can be simplified more so.