In turbojet engines, does the fan play a key role in the air intake to the inlet?
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In turbojet there is no fan. Very simply, turbofan means turbojet + fan in front connected to the LP shaft.
Further, nothing about a turbofan or turboprop requires the fan/prop on the front. It absolutely can be set up in pusher mode, with the prop or fan behind the engine, and turbine exhaust being diffused into the fan input. If anything, this is easier, because it means the prop/fan is on the same end of the engine as the second stage (low pressure) compressor, so there's no need for that shaft to run all the way through the engine. Thus reducing the need for concentric shafts.
So in that case, the engine core will input air the same way a turbojet does. There may be a second input for bypass air.