Mainly Need help with the math:

The goal is to launch(NOT levitate) an object that is 3x the weight of the electromagnet 20cm high.

I'd like to use the 6000mAh lipo battery I already have if possible (so 6Amps)

I found 7 AWG square magnet wire, that’s the thickest I found in small consumer size quantities.

diameter: 0.144in which is 0.0036576m

weight: 15.87ft of this wire weighs 1lb

reference: https://mwswire.com/wp-content/uploads/2016/10/Copper-Magnet-Wire-Data.pdf

For the core; 5mm diameter and made of closest to pure iron I can get or better/easier to get.

reference: see section on 'Values for some common materials' https://en.wikipedia.org/wiki/Permeability_(electromagnetism)#Values_for_some_common_materials

This will help with the magnetic permeability variable I've seen in electromagnetic field strength equations.

With these above set values I believe the last thing I would need to know is the number of turns in the coil and the length of the "gap" - with my best guess means the length of the core? or how many levels in the coil(i hope Im describing that correctly).

It seems I'll have to guess and check from there, i.e. the weight will be determined by the number of turns and length the gap.

What equations do I use?

I found this site for calculating the length of wire: https://www.daycounter.com/Calculators/Coil-Physical-Properties-Calculator.phtml

Which will be helpful in determining the weight of the coil.

For electromagnetic field strength equations;

I found this video which I think is wrong:

Then there is the question of the force T(teslas) how much force is even needed to launch(NOT levitate) 3x the weight of the electromagnet 20cm - I do not know where to find an equation for this.

AND if 6Amps isn't enough - than what battery?