I suggest reading this article which includes a mathematical formula for determining the habitable range around a main sequence star.
Basically, start of habitable zone (least distance) = $\sqrt {L_{star}\over 1.1}$ AU
AU = astronomical unit (150 million km)
$L_{star}$ is known as Absolute Luminosity of the star.
The formula for Absolute Luminosity is: $L_{star}$ = $_{10} \left[{M_{bol} - 4.5}\over{-2.5} \right]$
$M_{bol}$ is known as Bolomatric Magnitude of the star. In order to calculate it, you use this formula:
$M_{bol}$ = $M_v + BC$
Here $M_v$ is known as Absolute Magnitude of the star. $BC$ is Bolometric Correction constant. It depends on the stellar class of the star. There are 6 stellar classes, B A F G K M. Their bolometric correction constants are -2.0, -0.3, -0.15, -0.4, -0.8, -2.0 respectively.
In order to calculate $M_v$ (Absolute Magnitude), you use this formula:
$M_v$ = $m_v - 5$ x $log{d\over10}$
Here $m_v$ is the Apparent Visual Magnitude of the star (aka its visual spectrum). $d$ is the distance of Earth (our planet) from the star. This distance is measured in parsecs (one parsec is nearly 3.26 light years).
In Short
You would need to know these things if you want to calculate the habitable around a star:
- its stellar class
- its apparent visual magnitude
- distance between Earth and that star
My Advice
This site is about world building. While it is good and highly recommended to keep your world building compatible with real world values, scientific research and world building are two distinct things.
In order to set up a habitable zone around your star, I would suggest reading this wikipedia article. There is a table of stars and their habitable zone distances. You should see that and then make an estimate for the habitable zone around your star. Common sense and pattern recognition should land you fairly close to the real values for your star.