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.