I guess 4A you mean 4 ampere / hour
Batteries manufacturer declaration on A/h are very fuzzy and depend on the current consumption and maximum battery voltage drop allowed
The best way is to test
If this is 3V LED, the voltage regulator is set to 3.1V and 286Ω resistor is connected in series with the LED then something is wrong, as this combination will allow only 0.35mA through the LED ..
Better configure the LM317 as 350mA constant current source and don't use any resistor in series with the LED ..
If your battery is rated for 4Ah then, in theory, you can draw 350mA for (6Ah/0.35A) 17h ..