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Your starting system should always have a yellow star with a luminosity of ~1 and at this point most of your industry is probably there, so for starting out you can make a sphere in that system.
Btw. if you're lucky the closest planet to the star is inside the maximum sphere diameter.
Build your ray receivers on a planet inside the sphere and they'll always have 100% up-time.
Since the ray receivers on the sun side will always be facing and receiving. Plus the rail guns for solar sails will be able to constantly fire.
They also can make good accumulator charging planets, especially if they have a good % solar power bonus.
You can't actually run out of oil, just the tapped rate drops to a very low base rate.
There are few solutions to fixing though,
1. get more veins utilization research to bump up the base rate. Also reduces the rate that the wells reduce production at
2. go to other worlds with warpers and tap those oil wells and ship the oil back. Since oil the decrease only happens when you are actively pumping
3. If you are not already using them, start using alternatives production chains to reduce your oil use. The big two being Ash worlds that have Sulfuric acid you can pump, and fire ice (which some gas giants give, so it is an unlimited rare resource) for Graphene.
Fire ice mainly reduces oil use indirectly, assuming you have not yet found an ash world with Sulfuric acid oceans
this may have changed i dunno for sure, my new system i started fresh because i paved over my magma world and i wanted to mess with the geothermal power. they are worth it and power everything i need for production currently all on their own.
finding an O type star and sending the materials out there to get one built or started so you can power everything there, then use that power to ferry resources back to your start system to make more dyson sphere is honestly a better option.
As for the stars themselves. There's two factors that determine the usefulness of a star for building. The main factor, and most direct one, is 'Luminosity'. While O type stars tend to have the highest luminosity - there is some overlap with the B spectral class. Occasionally the seed for your game will spit out O types with abnormally low luminosity and a B type with unusually high luminosity such that the B type ends up the highest in your cluster. Pretty much anything close to or over 2-3 L range will serve your purposes, no matter if it's O type or B type, it's just that O types will have the highest luminosity - on average.
This brings us to the second factor, the size of the star. The size of a star has no direct impact on energy output. Two stars with 2.5L, one a small star and one a giant, will produce the same output when ringed with the same size Dyson shell. The benefit comes in that with a larger size star, there is just physically more space to build a larger shell. So you can draw more power from a single larger star than a single smaller star, but you will need to build that larger infrastructure in order to do so. Blue giants can be of A, B, or O spectral class, the very subtle shading in-game will let you know which class your giants are - O types are a darker shade of blue than B types, and A's will be pretty obvious as to be almost white with only a hint of blue-ness (thankfully A class Blue giants are fairly rare - you're much more likely to find a class A White giant before you find a Class A Blue giant) - this also mirrors the color shading of regular stars themselves.
You can build a 10k point Dyson frame around a small 2.5L star (which might be the max size for it), and a 10k point frame around a giant 2.5L star to get X total amount of power from both. Or you can build one 20k point shell around the giant 2.5L star (because it has plenty of room spare) to get that same X amount of power. At which point, it's personal preference. As the old expression goes "Six to one; half-dozen to the other"
Sure, they are more luminous, but they are larger, so take more effort to build - doesn't that cancel out?
Example: in my new game I have a yellow star nearby that is roughly 1 luminosity and 1 radius. Closest blue star is about 1.8 luminosity but 2.6 radius. So it is 80% brighter but 260% more expensive to build.
Doesn't that make the blue star worse than the yellow star?
Also the blue star's system is larger and getting from the inner plant to the outer plant seems to require a gravitron lense unlike the yellow star's system where I can just sail between the planets.
Blue is better because it's actually cheaper to build for each watt of power you generate. It's also because people are talking about maximum power output. Yes, smaller less luminous stars work just fine and you can build smaller and save parts, but you won't get as much power. If you build the same size sphere around each one the blue will put out a lot more power with the same number of components. I always build my first spheres around my home stars which gives a very good start. The blue stars come in later when you are trying to generate obscene amounts of power.
It doesn't cancel out.
Each sphere component like solar sails or nodes produce power.
The more components a sphere contains the more power it produces.
Since O type stars are pretty big the spheres you can build around them are also larger with more space for components.
Luminosity on the other hand is a modifier - it increases the power output for each component.
If you want a complete Dyson Sphere layer with the lowest resource investment then a smaller star, like a yellow one, is better.
If you're looking for the maximum power for a given resource investment then a max luminosity star is best (regardless of size). A 2.5 luminosity star will produce 2.5x as much power from each small rocket or solar sail as a 1 luminosity star -- even if the Dyson Sphere is incomplete.
If you're looking for the maximum total power output from a Dyson Sphere then you want both high luminosity and large stars. Building all 10 layers, each as large as you can, around such a star can yield north of 10 TW of power! But completing such a monstrosity does take an extreme amount of rime and resources.
Generally when people ask about the "best" star for a Dyson Sphere the assumption is they're asking about the 3rd goal -- max power after completion -- and thus the recommendations for huge high luminosity stars. But if that's not your goal then adjust your preferred star(s) accordingly.