Easy circuit for connecting LEDs directly to AC mains without a DC power supply. The RMS current through the circuit can be approximated by Vrms/sqrt(R²+1/(2pi*f*C)²), since diode forward voltage is small compared to AC mains voltage. For a single LED, connect a normal diode in anti-parallel config (diode cathode goes with LED anode and vice-versa), in order to avoid DC currents in power lines, which can be detrimental to them.
