Minimum Difference Function

For any positive integer n let f(n) denote the minimum difference
|a-b| for any two integers a,b such that ab=N.  Then define F(n)
as the sum of f(k) for k=1,2,..,n.

When is F(n) divisible by n?  Here's a table of all the occurrences
for n < 175000:

                 n         F(n)       F(n)/n
              ------     --------    --------
                   1           0         0
                   3           3         1
                   7          14         2
                   8          16         2
                  55         550        10
                  75        1050        14
                 146        3504        24
                 204        6732        33
                 224        7840        35
                 679       63826        94
                 831       93072       112
                 860       98900       115
               63057   328085571      5203
              113740  1009897460      8879
              114507  1022891031      8933

Are there infinitely many such occurrences?

I've subsequently found a couple more, so all such n I know are

    1, 3, 7, 8, 55, 75, 146, 204, 224, 679, 831, 860, 63057, 
    113740, 114507, 660479, 2329170, ....?

This raises a more general question.  Suppose f(n) is defined as 
a randomly selected integer in the range from 0 to n, and the 
cumulative form F(n) is the sum of f(k) for k=1 to n.  What is 
the expected density of n values that divide F(n)?

Robert Israel notes that if we made the range for f(n) be 0 to n-1
(instead of 0 to n) then F(n) = F(n-1) + f(n), and for any F(n-1) 
there's exactly one f(n) value that would make F(n) divisible by n.  
So the probability that n divides F(n) is exactly 1/n.  With the 
range 0 to n, he remarks that the exact answer is more complicated, 
but presumably very close to 1/n.