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I extended the search to $n \leq 111111$ and find this: $$ \frac{10090}{110990} + \frac{36997}{110991} + \frac{15856}{110992} + \frac{6529}{110993} + \frac{8538}{110994} + \frac{22199}{110995} + \frac{55498}{110996} + \frac{36999}{110997} + \frac{55499}{110998} + \frac{95142}{110999} + \frac{55500}{111000} + \frac{100910}{111001} + \frac{55501}{111002} + \frac{74002}{111003} + \frac{27751}{111004} + \frac{88804}{111005} + \frac{55503}{111006} + \frac{102468}{111007} + \frac{27752}{111008} + \frac{74006}{111009} + \frac{104480}{111010} = 10. $$