Is there a mathematical formula for electronegativity? If we lost all the numerical values we have for the electronegativity of elements, is there any mathematical formula that can be used to re-calculate them based on proton (and neutron) numbers alone? For example, if I had no way to look up the electronegativity of Carbon, could I use some existing formula to calculate it from scratch? Or would I have to go actually measure it?
Electronegativity cannot be calculated from the number of subatomic particles in an atom. However, using Ionization Energy and Electron Affinity, one can calculate the Electronegativity of an atom using the Mulliken Electronegativity Equation.

Ionization Energy : The amount of energy required to remove the
electron that is least attracted to the nucleus in a gaseous state.
Electron Affinity : The amount of energy an atom gains when an
electron is added.

To find the Ionization and Electron Affinity values of an atom, one must reference a chart.  In the Mulliken Electronegativity Equation, Ei stands for Ionization Energy and Eea stands for Electron Affinity.  Plug in your values and solve.

Equation for Energies in Kilojoules/Mole :
ENMulliken = (1.97×10^−3)(Ei+Eea) + 0.19
Equation for Energies in Electronvolts :
ENMulliken = 0.187(Ei+Eea) + 0.17

Lithium Example in kJ/mol :

The Ionization Energy for Lithium is 520 kJ/mol
The Electron Affinity for Lithium is 60 kJ mol^-1
ENMulliken = (1.97×10^−3)(Ei+Eea) + 0.19
ENMulliken = (1.97×10^−3)(520 + 60) + 0.19
ENMulliken = 1.143 + 0.19 = 1.333


NOTE : There are multiple scales for measuring Electronegativity, including the Pauling Scale, Sanderson, Allen, and more.  This example measures Electronegativity using the Mulliken scale, and will compute an answer that will differ from other models.