Here is some computational evidence for finiteness of such $n$. For an odd prime, let $a_p < b_p$ denote two smallest even positive integers such that $a_pp+1$ and $b_pp+1$ are prime. Then $n\in [a_pp+1,\min(b_pp+1,p^2))$ cannot deliver an integer sum as its denominator contains $p$. Here are these intervals for primes below $100$:

    3 [7, 9)
    5 [11, 61)
    7 [29, 211)
    11 [23, 331)
    13 [53, 859)
    17 [103, 1871)
    19 [191, 3877)
    23 [47, 1151)
    29 [59, 1741)
    31 [311, 9859)
    37 [149, 6143)
    41 [83, 3527)
    43 [173, 7741)
    47 [283, 13537)
    53 [107, 6043)
    59 [709, 42953)
    61 [367, 22571)
    67 [269, 18493)
    71 [569, 40471)
    73 [293, 22193)
    79 [317, 25439)
    83 [167, 15107)
    89 [179, 17623)
    97 [389, 37831)

As we can see primes 5, 23, 59 alone cover the interval $[11, 42953)$, while others provide extensive backup support. Such a pattern will likely continue to cover all integers above $11$. However I'm not sure whether this observation can be justified theoretically without relying on unproved conjectures.