Question
I want to do core aeration and over seeding.

Can I do one immediately after the other and which do I do first?  

Answer
Good afternoon Leo:

For successful overseeding, it is critical that seeds be placed into contact with the soil, whether seeding into live or dead turf. Seeds scattered on top of live or dead turf rarely produce a decent stand of turf. Several ways exist to place seed in contact with soil.
To open the soil for seeding in large areas, use gas-powered vertical mowers,
slicers, core aerifiers, spikers, or slit-seeders.

Core aerifiers open soil by inserting hollow tines into the ground and removing
plugs of soil. When using a core aerifier for overseeding, it is important that at least 20 plugs per square foot be extracted. After broadcast seeding, allow the plugs to dry and use a drag (a piece of chain link fencing works well) to work seeds into the soil and crumble the plugs.

Another method you might consider is a slit-seeder. These are recommended when you presently do not have a thatch or compaction problem.
Slit-seeders (mechanical overseeders) are considered to be the most convenient method of placing seeds in contact with soil. These machines combine a vertical mower with a seed spreader梩hey open the soil with a vertical mower mechanism, and then seeds are metered at a predetermined rate from a holding hopper through feeder tubes and dropped into the slits made by the vertical mower. Machines often have a roller that firms the soil after seeding. Slit seed the area twice; the second pass should be at a 45?angle to the first pass.

To renovate successfully, adequate water must be available from seeding through completion of germination. Try not to allow newly overseeded areas to dry once watering has commenced. Initially, maintain a constantly moist seedbed with frequent waterings of short duration. As newly germinated seedlings begin growth, decrease watering frequency and increase the duration to encourage rooting.

I hope I have answered your question.
Good Luck!
Floyd McMahon