In photography, what is interesting is mostly the angle of view (AOV). The AOV is the angle that a lens offers on a sensor - it can be specified horizontally, diagonally, or vertically.
AOV [°] = 2 * arctan ( sensor_height|width|diagonale [mm] / (2 * focal_length [mm]) )
The formula to get from a specified focal length (FL) on a non-full-frame sensor to the full-frame-equivalent focal length is:
equivalent_FL [mm] = true_FL [mm] * crop_factor
The crop factor can be determined by comparing the diagonals:
crop_factor = full_frame_diag [mm] / your_sensor_diag [mm]
- With the same focal length, a larger sensor (but same aspect ratio) will give a greater AOV
- With the same sensor dimensions, a smaller focal length will give a greater AOV
- AOV is different in vertical, horizontal, and diagonal (except in a quadratic sensor, where vert&hor would be the same) axes
Or, in practical terms:
- A 10mm lens on your 5.6-crop-factor sensor will give you an AOV that is equivalent to that of a 56mm lens on a full frame sensor.
- The same 10mm lens on a 1.6-crop-factor sensor will give you an AOV that is equivalent to that of a 16mm lens on a full frame sensor.
- A 1600mm lens on a full frame sensor will give the same focal length as a 1000mm lens on APS-C (1.6 crop) or a ~ 285mm lens on your point and shoot.
- A 16mm lens on a full frame sensor will give the same focal length as a 10mm lens on APS-C or a ~ 2.85mm lens on your point and shoot.
- All other factors left aside, smaller sensors favor smaller AOVs /
higher reach, while larger sensors favor wider AOVs.
- Among the ignored factors are:
- Pixel density (a 20mm² sensor with 20MP has half as large pixels than
a 40mm² sensor with 20MP) which influences noise (smaller pixels
typically are worse at collecting light and thus contain more noise)
- Aperture (f/4 on a 5.6-crop-factor is something like f/24 on full
- Physical limitations (e.g. negative-valued focal lengths
(-1mm) are not possible)
Why do we then use focal lengths (in mm) on lenses? Because AOV is not a function of the lens, but of the sensor-lens combination. A lens will keep its focal length forever, but based on the sensor it is mounted on, its AOV will vary. (Of course, the image circle that a lens can provide will limit its abilities at some point, so mounting a smartphone 3mm lens on a medium format sensor wouldn't do much good ;-) )
Oh, and why compare it to full frame? Because we needed some metric to compare it to - we could also use IMAX or Super35 or
1 / (⅔ * π) [inches] if we want to.
Now to actually answering the question:
Your formula was:
(1365 / 5.6) * 1.6 = 390
Which would mean:
effective_FL / crop_factor_PnS = real_FL_PnS
real_FL_PnS * crop_factor_APS-C = ??
What you calculate is therefore the effective focal length of the lens of your point and shoot camera on the sensor of your new camera.
Your 1365mm are already full-frame equivalent, so you can calculate the APS-C related true focal length with this value already.
1365 / 1.6 = 853.125 [mm]
So you would need a lens with that focal length to get the same narrow AOV with a 1.6-crop-factor sensor.
Note that the difference in AOV between 100-200mm is bigger than that between 500-600mm!
Note that - as twalberg already stated - 400mm+ lenses are usually very expensive and mostly limited to primes (and/or the use of tele-converters, which might deactivate your camera's AF if your lens is not fast enough). This is because they are typically a niche market built for professionals who need/want every last bit of image quality, and most 15000€ lenses on 5000€ bodies offer better image quality in the worst of circumstances than any 500€ camera ever can. Does that mean it will make you a better photographer or that you need that setup? No!
I have no stake in this, but if you want a modular system with that kind of reach, I think that µ4/3 might be a better choice if you are on a budget - it offers a 2x crop and 100-400mm lenses are not quite as costly as an 800mm prime from Canon ;-)