When we connected LEDs to AC transformers, the calculation we used was:
Vac / 3.3
to give us the number of LEDs we need to be able to properly handle the power without additional resistors and other parts.
What if we bypass the transformer completely and consider AC mains? In some ways it is simpler - the voltage from transformers could vary greatly with the load we put on it, whereas AC mains are much more stable.
If we use the 110v standard of the US, we first calculate the peak voltage, 1.4 * 110 = 156 and we can find the number of LEDs it can support:
156 / 3.3 = 47 LEDs
So, does that mean that if we put 47 LEDs in series, we can run the whole string directly off a 110v AC socket?
The answer is Yes! As long as we maintain the voltage across each LED at 3.5v or less, it will operate within its limits.
But then, let's not forget that for each positive cycle, there is a negative cycle! That means we need a mirror circuit like in (1).
Wow, that's an awful lot of bulbs!
However, if we add a blocking diode like in circuit (2), then we can safely operate our circuit. The 1N4003 is capable of handling 200 volts so is fine for US power.
For EU countries, the magic number is 103 LEDs (double if you want to use both cycles) and the diode for ckt (2) should be a 1N4004 or better.