Estimating the fleet size from six serial numbers

Vehicles in a fleet carry distinct serial numbers $1,2,\dots,N$, with $N$ unknown. You see six of them at random, numbered $38, 17, 59, 42, 97$ and $120$. Using a frequentist (unbiased) estimator, what is your best estimate of $N$?

$139$. With sample maximum $m=120$ and $k=6$ observations, the gaps below the maximum are, on average, equal: the $114$ unseen numbers below $120$ are split into $6$ gaps of about $19$, so one expects another $19$ above the maximum. Equivalently the minimum-variance unbiased estimator is $m+\tfrac{m}{k}-1=120+20-1=139$.