Here is some computational evidence that $n\in\{1,2,4\}$ are the only $n$ that deliver an integer sum. 

For an odd prime $p$, let $a_p < b_p$ denote two smallest even positive integers such that $a_pp+1$ and $b_pp+1$ are prime. Then $n\in \big[a_pp+1,\min\{b_pp+1,p^2\}\big)$ cannot deliver an integer sum as its denominator will necessarily contain $p$. Here are these intervals for primes $p<100$:

    3 [7, 9)
    5 [11, 25)
    7 [29, 43)
    11 [23, 67)
    13 [53, 79)
    17 [103, 137)
    19 [191, 229)
    23 [47, 139)
    29 [59, 233)
    31 [311, 373)
    37 [149, 223)
    41 [83, 739)
    43 [173, 431)
    47 [283, 659)
    53 [107, 743)
    59 [709, 827)
    61 [367, 733)
    67 [269, 1609)
    71 [569, 853)
    73 [293, 439)
    79 [317, 1423)
    83 [167, 499)
    89 [179, 1069)
    97 [389, 971)

As we can see primes 5, 11, 29, 41, 67 alone cover the interval $[11, 1609)$, 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.

**ADDED.** Primes 5, 11, 29, 41, 67, 743, 7823, 165293, 6215171 cover the interval $[11,1131161123)$.