The number is close to the ratio of the area of the large circle divided by the area of the hexagon the small circle will fit in, which is pi R^2/(2sqrt(3)r^2).

This is more accurate for larger numbers but may bee slightly too large. A safer number takes away the number of small circles that fit around the perimeter, so is

pi R^2/(2sqrt(3)r^2) - pi R/r, which is pi R/r (1/(2sqrt(3)) R/r - 1)

I think this tool might be helpful:

https://www.engineeringtoolbox.com/smaller-circles-in-larger-circle-d_1849.html

If you only take away half the area of the circles that will fit around the perimeter, then this is close to the result of the website Vincenzo Antonio Rossi recommends.

pi R^2/(2sqrt(3)r^2) - pi R/(2r), which is pi R/(2r) (1/(sqrt(3)) R/r - 1)