In one of my current works in slow progress, I debated this number issue and came up with what I hope works to START a colony.
1...The planet has to be able to support life, water, air, and basic food products (edible plants and or animals), have to already be there and our genetics have to tolerate this new food.
This new world would also have the basic building elements in order for a new colony to survive. Trees for lumber to build houses and use for fuel, furniture, and basic equipment like plows and carts. Also, coal, iron and other metals to be mined to make crude instruments for mining, building, farming, etc. Crude forges could be built fairly early, but the minerals have to be there and somewhat easy to get to.
2...Because it will take a large gene pool for any colony to survive without developing genetic DNA line problems, my colonist have more women than men. Everyone understands that monogamy is out for several hundred years so the colony line can be established. Of course, saying monogamy is out and actual practice will be difficult and no will be forced into relationships they do not agree with.
3...Obviously not everyone involved is going to be a specialist of some sort. The colony will need not just the scientist, doctors, veterinarians, engineers, technicians and so forth, but will also need specialized skills in farming, fishing, building anything from houses to boats, craftsmen to work in metals, and even specialized laborers for building and food production. Some type of military or police members will also be needed just to keep the peace in the colony as well as protect the colony.
4...Politics. Yes there will be politics, but before the ships ever leave earth, a set of laws for the colony will have to have been drawn and everyone will have had to agree to them. Once the colony is established in twenty years, then new laws can be written, added, or even old laws removed. A single leader will be needed for the first five years and then the colony can vote on a new leader and elders or senators to represent them. As new villages form, leaders in those villages will play into the political scene as they represent their village needs to the main colony city. If that makes sense...
5...What I came up with is 15,000 colony starters, minimum, with no children. All adults. The children can come in a few years after the basic colony has been established. The first generation should more than double the colony in the first twenty years, excluding any massive plagues or die offs. The Second generation would then double again in another 20 years and the colony could start spreading out and establish additional towns and villages. By the fourth generation, 80 years after the first generation arrived, trade between the different towns and village will start progressing and we end up with the start of a new world as each generation spreads out.
This is all based on limited interaction between the mother planet and the colonies as supplies will be costly and take a long time to get. So the colony has to be self sufficient and that means basic fuels and energy is going to be limited until what they brought is gone. Sure, we can bring a lot of solar stuff, but when that stuff starts breaking down, you have to have the ability to repair or replace parts which a new colony will not have. Paper will once again become a necessity because tablets and lap tops will break down, or just become useless with the power to run them. Same with all electronics unless you want your colony to develop nuclear energy, or some other such energy form. Wind of course would be a blessing for small scale energy.
I see my colony almost starting out much like Europe did during the first century. My colonist for all purposes would start out in the first century of their new world and a lot of exploring will occur, especially once they realize they are not alone...
Now, if the colony is on a world with limited natural resources, then you will have to figure out what is the value of settling on a world that has limited resources. A mining colony would be far different from a colony sent to populate a new world.
I hope some of this makes sense...