There's uncertainty introduced by the idea that you don't know how big that range is supposed to be. Technically the M/n figure should be (M-L)/n, where L is the sample minimum observed, and the estimated population minimum would be represented by L + (M-L)/n - 2 [1].
[1] two degrees of freedom should be lost due to estimating both M and L, thus subtracting 2.
It does seem to be a harder problem if are given that they do not start at 1, but that they are consecutive integers. I imagine you might do a similar sort of statistical estimate on the beginning of the range as well as the end of the range. This introduces more uncertainty, but I see no reason why this wouldn't work