I extended the search to $n \leq 111111$ and find this: $$ \frac{10090}{110990} + \frac{36997}{110991} + \frac{15856}{110992} + \frac{6529}{110993} + \frac{8538}{110994} + \frac{22199}{110995} + \frac{55498}{110996} + \frac{36999}{110997} + \frac{55499}{110998} + \frac{95142}{110999} + \frac{55500}{111000} + \frac{100910}{111001} + \frac{55501}{111002} + \frac{74002}{111003} + \frac{27751}{111004} + \frac{88804}{111005} + \frac{55503}{111006} + \frac{102468}{111007} + \frac{27752}{111008} + \frac{74006}{111009} + \frac{104480}{111010} = 10. $$
In case you want to verify this more efficiently, here is the "code version":
10090/110990 + 36997/110991 + 15856/110992 + 6529/110993 + 8538/110994 + 22199/110995 + 55498/110996 + 36999/110997 + 55499/110998 + 95142/110999 + 55500/111000 + 100910/111001 + 55501/111002 + 74002/111003 + 27751/111004 + 88804/111005 + 55503/111006 + 102468/111007 + 27752/111008 + 74006/111009 + 104480/111010
EDIT:
Now that almost one year passed, I would like to confess that the "extended the search to $n \leq 111111$" part was a joke...
Sadly nobody seems to find it funny ...
Or perhaps nobody ever cares (including the OP, who is probably busy conjecturing all kinds of things) ...