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Sum of Sums over Primes that Divide the Index

  1. Sep 25, 2012 #1
    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:

    [itex] \sum_{k=1}^{n} \sum_{p | k} \frac 1p [/itex]

    for p prime.

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

  2. jcsd
  3. Sep 25, 2012 #2
    I think that's equivalent to [itex] \sum_{p=2}^{n} \frac{\left \lfloor n/p \right \rfloor}{p} [/itex], 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.
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