As far as I Know, range will be increase when flying on one engine only if we HAVE TO fly at very low altitude e.g. PRESSURIZATION FAILURE.
In a twin MD 80 flying at FL 130 at LRC and GW 54T Specific Range is 117 nm/t of fuel on 2-eng, while with one engine shut-down Sr increase to 121 nm/t, not much, but a little increase.
At higher FL, 2 engine specific range is higher.
At low altitudes, when 2 engines are runnig for LRC, power settings are really very low, therefore engines are very inefficient, if you shut down one engine, remaining one will increase N1 and will be more efficient.
I donīt know about DC-8, but I guess that at low GW and FLs up to 250 the same may apply when flying on two engines instead of 4.
REGARDS