General number field sieve is slower then exhaustive search for 'small' numbers? In an attempt to understand the efficiency of the GNFS, I've been looking at runtimes.  The calculations seem to indicate the GNFS runs slower than exhaustive search for smallish n.  For example: suppose I want to factor the number 247 (ie 13 * 19).

To do this I can just check all the numbers (exhaustive search) below 
$$ \sqrt{247} \approx 16$$ to see if they divide 247.

The General number field sieve has a runtime of 
$$
\approx \exp \sqrt[3]{\frac{64}{9}} (\ln n)^{1/3} (\ln \ln n)^{2/3},
$$

So calculating the GNFS runtime...
$$
\approx \exp \sqrt[3]{\frac{64}{9}} (\ln 247)^{1/3} (\ln \ln 247)^{2/3},
$$
$$
\approx \exp \sqrt[3]{\frac{64}{9}} (5.5)^{1/3} (\ln 5.5)^{2/3},
$$
$$
\approx \exp \sqrt[3]{\frac{64}{9}} (5.5)^{1/3} (1.7)^{2/3},
$$
$$
\approx \exp (1.9 * 1.8 * 1.4),
$$
$$
\approx \exp (4.87) \approx 120
$$

This is obviously bigger than 16.  What am I missing here?
First of all, the number 120 that you got is completely meaningless. There is a scaling factor that can only be determined experimentally. Even worse, there are lower order factors which are probably very significant for such small numbers. Finally, the running time you quote is heuristic – it's just an educated guess.

You are correct that GNFS is a lot less efficient than trivial methods for small numbers. That's why you don't use it for small numbers. The better asymptotics kick in only for large enough numbers.