Lets says there is a planet in the goldilocks zone of a type A star (luminosity of 16 and mass of 1.9 (like the star Fomalhaut))
How far would the type A star need to be from the planet and how would that look in the sky related to the sun size and brightness?
Your queston should ask how far your planet should be from the star instaad of vice versa, since the planet orbits around the star.
The Sun has an absolute magnitude of 4.83, which thus is equal to 1 solar luminiosity. Note that magnitude is on a logarithic scale and that the lower the magnitude, the more luminous the sta, and the higher the magnitude the less luminous the star.
The Earth's ellipitcal orbit means that it is sometimes closer and sometimes farther from the Sun. But the semi-major axis of the orbit of the Earth around is Sun is approximately one Astronomical Unit or AU. The new definition of an AU is 149,597,870.700 kilometers.
Since the Earth orbits around the Sun, which has 1 luminosity, and recieves the amount it recieves from the Sun at a distance of 1 AU, for a star of the same spectral type as the Sun a distance of 1 AU is the distance at which a planet will receive as much radiation as Earth gets from the Sun. I call that distance the Earth Equivalent Distance, or EED.
So what you are asking for is the EED of a class A star with a luminosity of 16 times the Solar luminosity and 1.9 the mass of the Sun, like Fomalhaut.
Fomalhaut, or Alpha Piscis Austrini, is a triple star. The main star, Fomalhaut A, is class A3V, has 16.63 plus or minus 0.48 solar luminosity, 1.92 plus or minus 0.02 solar mass, and has a radius of 1.842 the radius of the Sun.
You might assume that if Fomalhaut A has 16.63 times the luminosity of the Sun, its EED would be 16.63 AUu from the star. But the intensity of radiation received from a source varies with the square of the distance from that source. Thus the EED of Fomalhaut A should be at a distance of the square root of 16.63 AU, or 4.0779897 AU, or about 610,058,575.9 kilometers, which should be the approximate semi-major axis of the planet's orbit.
If Fomalhaut A has 1.842 the radius of the Sun, and if the planet is 4.0779897 times as far from Fomalhaut A, Fomalhaut A should appear about 0.4516931 as wide from the surface of the planet, as the Sun appears from Earth.
According to the chart at this article: https://en.wikipedia.org/wiki/A-type_main-sequence_star#:~:text=An%20A%2Dtype%20main%2Dsequence,strong%20hydrogen%20Balmer%20absorption%20lines.
A spectral class A3V star should have a mass of 1.89 solar mass, a radius of 1.861 solar radius, and a luminosity of 16.98 solar radius. And those values for a typical A3V are a little different than those of the specific A3V star Fomalhaut A, and no doubt a professional astrophyscist could explain why.
Going with the values for a class A3V star, you find that the EED will have a distance of 4.1206 Au, and that the class A3V star will have an apparent diameter of about 0.4513 tha tof he Sun as seen from Earth.
You actually didn't ask for the exact EED distance between the planet and the star. You just asked for a distance within the circumstellar habitable zone of a star. And if the luminosity of a star relative to the Sun is known, it is simple to calculate the inner and outer edges of a star's habitable zone.
Except that the inner and outer edges of the Sun's circumstellar habitable zone aren't known with any precision.
Here is a link to a list of about a dozen estimations and calculations of the inner or outer edges, or both, of the Sun's circumstellar habitable zone.
https://en.wikipedia.org/wiki/Circumstellar_habitable_zone#Solar_System_estimates
Note the great difference in how wide orhow narrow those estimates make the Sun's circumstellar habitable zone. So my advice to writers writing a story where a star sysem has one and only one habitable planet is to put that planet at exactly the EED distance for that star, or within one or two percent of the EED, to be certain that the planet will be in that star's circumstellar habiable zone.
If someone writes a story where a star system has two or more habitable planets in different orbits around the star, they will have to assume that the star has a broad enough circumstellar habitable zone to have two or more planetary orbits within that zone.
You should know the diference between a planet which is habitable for liquid water using life in general, and for humans (and beings with similar requirements) in particular.
Even today, there are many parts of the bioshphere of Earth where many lifeforms flourish but unprotected and unprepared humans would swiftly die. And it took billions of years for Earth's oxygen rich atmosphere to form (produced by lifeforms which didn't need oxygen) and for Earth to become habitable for humans.
Most scientici discussion of planetary habitability discuss the general cases of habitability for liquid water using organisms in general. As far as I know the main scientific discussion of habitabity for humans (and similar lifeforms) is Habitable Planets for Man, Stephen H. Dole, 1964.
https://www.rand.org/content/dam/rand/pubs/commercial_books/2007/RAND_CB179-1.pdf
On pages 61 to 63 Dole discusses the minimum age for a planet to become habitable for humans.
In general,it is probably safe to say that a planet must have existed for 2 or three billion years, under fairly steady conditins of solar radiation, before it has matured enough to be habitable.
Note that Dole says the planet has to have fairly steady conditions of solar radiation for those 2 or 3 billion years. If there are large changes in the star's luminosity, it might kill all life on the planet.
A star can shine fairly steadily for at least 2 or 3 billion years while it is in the main sequence phase of its development. But in other phases its luminosity will change drastically, and will not stay the same luminosity long enough for any planet to become habitable for humans. It is true that a white dwarf star will also shine fairly steadily for many billions of years, but it will be so dim that only planets very close to it could be warm enough for liquid water life, and any such planets would have been destroyed when that star was in the red giant phase.
So naturally Dole discusses what types of stars could remain in the main sequence stage long enough to become habitable, on pages 67 to 72. The more massive a star is, the faster it will go through the main sequence stage and become a red giant, destroying all life on any habitable planets it might have. On page 72 Dole concludes that:
The range in mass of stars that could have habitable planets is thus 0.72 to 1.43 solar masses, corresponding to main sequence stars of spectral types F2 though K1.
So that would seem to leave out all O, B, and A class stars from consideration as having habitable planets orbiting around them.
According to Wikipedia:
A-type stars are young (typically few hundred million years old) and many emit infrared (IR) radiation beyond what would be expected from the star alone.
https://en.wikipedia.org/wiki/A-type_main-sequence_star#Planets
So even a class A star about to become a red giant might be only 10, 20, or 30 percent as old as the minimum age Dole estimated for a planet to becomehabitable for humans.
And of course it is statistically possible, I guess, for one class A3V start out of millions or billions to have a planet that naturally develops a breatheable atmosphere in a tiny fraction of the time that it took Earth to.
But it would be more plausible for a planet of a class A star to be terraformed by an advanced civiization to become habitable with a breatheable atmosphere. It might take the civilizantion centuries, or millennia, or much longer, to terraform the planet, but the planet might remain habitable for tens or hundreds of millions of years afterwards, and the civilization might consider the investment to terraform the planet worthwhile.
You might also want to see User177107's anwer to this question:
And this: http://www.pas.rochester.edu/~emamajek/EEM_dwarf_UBVIJHK_colors_Teff.txt
