Relative to the nation, what is the theoretical limit to how much a state's population can grow relative to the nation and still lose a seat in the US House?
The 2010 Census Briefs say the "The method of equal proportions has been used for apportionment after every census since 1940." It then goes on to detail the steps by which seats are apportioned:
- Automatically assign the first 50 seats, as every state is guaranteed at least 1 Representative
- Calculate a list of "priority values" using the formula PV(n) = (state apportionment population) / sqrt((next seat #) * (current seat #))
- Sort priority values in descending order and assign seats from largest priority value to smallest
The Census Bureau provides a handy Excel file of the priority values for the 2010 census here and the 2020 census here.
It's easiest to get large growth percentages if you start small, so let's see what happens if we start with the smallest possible state that can have 2 reps.
The priority value for the 435th seat in 2010 was 710231, so you'd need a population of ~1,004,419 to get that second seat instead of Minnesota getting its eighth.
Just to get some idea of a possible bound, let's assume zero national population growth and that residents spread themselves evenly across some number of states n. I think the priority value of the 435th seat can then be calculated using (US population / n) / sqrt(ceil(385 / n) * (ceil(385/ n) - 1)). This yields a plot with a peak at 39 states, for which the priority value of the 435th seat is ~835661.5. This implies a population of roughly 1,181,803 to barely miss a seat, so a hypothetical state could increase its population by roughly 18.1% and still lose a seat.
As a warning, I haven't done the calculations to see if this difference would increase or decrease if the national population changes, but it at least shows what kind of difference is possible.