Does anyone know what factor the magnets have to increase in size to account for not using solid cores? Just out of curiosity
that depends on how large the generator is and what your efficiency target is.
the largest 60hz transformers on the order of a gigawatt in capacity have leakage inductance that makes the impedance of the transformer drop 22% voltage under load. they are 99.8% efficient. my estimate is a transformer of 4 times the physical dimension (so 64 times the volume, which would make for an electrical capacity of 256 gigawatts assuming it could be cooled off), would operate more efficiently without a core.
the reason for this isn't really intuitive until its explained to you but basically the capacity of a motor or transformer scales faster than the volume if its possible to cool it off. a 50 megawatt transformer might weigh 50 tons, for a capacity of 1kilowatt per kilogram. and it would be better than 97% efficient.
a 97% efficient 1kw transformer will weigh about 10 kilograms, 5 kilograms if its a toroid (power density increases with flux squared, 1.3T for E core and 1.9T for good quality tape wound toroid)
the reason why is simply that the cross section area of the core sets the voltage per turn, and the cross sectional area of the core sets the amps. both increase with the square, and volts times amps makes for the 4th power, but the volume only increases to the cube.
so a 1 cubic inch 60hz transformer might be 50% efficient at reasonable power densities such as what's found in a "wall wart" and you might get 1 watt per cubic inch power capacity at more reasonable efficiencies, which is a power density of 7.8 watts per kilogram.
the magnetic core in a small motor let you wrap the magnetic flux around an electrical conductor of 100 times the cross section area that the flux lines would flow through air on their own. so the torque is set by the magnet (for the most part) but the increased magnetic path that the flux flows through, drops the copper losses perhaps 100 fold from what they would need to be to generate the same torque in the motor. i think it was tim williams (7 transistor labs) actually worked some real numbers out to illustrate a similar problem and he basically said if the resistance of metals such as copper were 1/200th what they are now, it would be practical to build air core motors.