S, just to get things clear.
You need to light 16 white LED's, which generally have a Vf for 3.5-3.7V. For decent brightness you'll need at thevery least 10mA, 15mA average...20mA would be pushing it for standard 5mm.
Available power voltage is 4.5V, probably 4.0 - 4.7 for 3 AA's.
Now, simply using resistors, it seems because of the limit of your power voltage, you would need a resistor for each LED, very wasteful. The value of which would be:
R =V/I, = (4.5-3.5)/ 0.015 (for 15mA) = 1.0/0.015 = 66ohm. = 68ohm standard value.
Altohugh I'm sure you are after 'simplicity' rather than having to venture into the world of switch-mode power supplies, I would strongly suggest either purchasing a small boost convert module, or building one yourself.
The power you require per LED is: 3.5 * 0.02 (for 20mA) = 70mW max.
For 16 LED's thats: 16 * 70mW = 1120mW = 1.12Watts.
Using a boost power supply would not only be more efficient, but also may negate the need for many resistors. A 'true DIY' boost power supply would be tricky to build, so I would look around for modules on ebay or something, or even buying a cheap 'LED flashlight', which generally have built in boost power supplies, but that would probably need to be modded in some fashion.
If you decide to use a boost power supply, then let me know and I'll see if I can find a cheap and convenient way of doing it. Either by using standard parts, or, more likely, buying a relatively cheap 'chip' which will do almsot all thework for you.
Blueteeth