I asked because I simulated this in ISIS and found your original calculations to be right for on one segment but all segments wouldn't light together. I built it with 7 LEDS instead and found that the reason it wouldn't work in ISIS is that the emitter current is being split between each branch while the collector current stays fixed. Each branch is getting just under 2mA for 7 LEDs.
I then found, that if I get rid of the base and collector resistors, and put a 150 Ohm resistor in series with the LED at the emitter, with 5V to the base, I get
(5 - 0.7 - 2) /150 = 15.3mA, thru each branch that I add, with the collector current increasing by around the same amount as I do.