LM338 based psu with arduino based voltmeter
The LM338 will regulate it's output so that it's 1.2V (the voltage you measured) higher than the adjust input. Ignoring the adjust input current for a moment you get a current of 

\$\dfrac{1.24V}{240\Omega} = 5.17mA \$

through the 240\$\Omega\$ resistor. The same current flows through the 3300\$\Omega\$ resistor, so the voltage across it will be 

\$ 3300\Omega \cdot 5.17mA = 17.05V \$

Adding the 1.24V reference voltage to that should give you \$18.3V\$ output voltage. You say you have 16V. Is your input voltage high enough? You need a 3V margin, so input should be 21.5V at least.  

edit
I ignored the adjust pin current because at 45 \$\mu\$A it's so much lower than the 5mA, but let's do the calculation properly:  

\$ V_{OUT} = V_{REF} \cdot \left(1 + \dfrac{R2}{R1}\right) + I_{ADJ} \cdot R2 \$
\$ V_{OUT} = 1.24V \cdot \left(1 + \dfrac{3300\Omega}{240\Omega} \right) + 45\mu A \cdot 3300\Omega = 18.44V
 \$  

so that agrees with our first calculation.