Is it safe to throw 3 week old and fully grown chickens together? No. It is possible you would be OK but it is a long way from safe. Is it safe for them to sleep in the same coop? Sure, if you keep them fenced in so they are separated. People tend to want absolute answers to what is safe but living animals don't work that way. There are always going to be risks and introducing 3 week olds to adults have several.
What is the safest way for you to introduce 3 week olds to adults? That depends a lot on what your facilities look like. We all have different conditions, what works for me might not work for you. Can you tell us the size, in feet or meters, of any coops and runs that you have? Maybe some photos or at least a verbal description of how they are tied together. A photo or two of the inside of your coop may be helpful. Do they free range or are they confined to a coop and run? How many chicks? The more information we have the more specific we can be in our recommendations.
There are generic suggestions but which you might be able to implement or how will depend on your unique situation. It helps a lot to house them behind wire for a week or more so they can see each other but not touch. When you let them mingle, separate feeding and watering stations can really help. At that different in age a panic room/safe haven may be possible. Having lots of places for the younger ones to hide under, behind, or above can really help. The more room you can give them the better off you are. Try to not force them to be confined in a small area together but let them work that out as much as they can on their own. Patience is often your friend.
Without knowing a lot more about your circumstances I can't get any more specific than this. People do it all the time, usually successfully. But occasionally you can have a disaster. I regularly integrate 5 week olds with my flock with no issues but I start working on that from their first day in the brooder, I have a lot of room, and I have a mature rooster in my flock. All those can help.