Note: Posted in MO since it is unanswered in MSE.
Analogous to Chebyshev's race between primes, I examined the race between primes in the greatest prime factors, GPF, of natural numbers. Similar to the regular prime race, in the GPF race, the proportion of prime of both forms is roughly $50\%$ each. However, unlike the regular prime race where the race is equal (A038691) or changes lead between primes of the form $4k+1$ and $4k+3$ (A096628) in the GPF race, $4k+3$ dominates. For $n \le 3 \times 10^{10}$, there are $14970975209$ numbers the GPF is of the form $4k+1$ and $15029024755$ numbers where $GPF$ is of the form $4k+3$ and up till this point, there is not a single instance where primes of the form $4k+1$ equal or take the lead in the race.
Let $f_1(n)$ and $f_3(n)$ be the number of natural numbers $\le n$ in which the GPF is of the form $4k+1$ and $4k+3$ respectively. The difference between the number of GPFs on these two forms appeared to increase in a very orderly manner as shown in the graph below.
Question 1: Does the GPF race ever become equal or change the lead i.e. is there an $n$ such that $f_1(n) = f_3(n)$?
Clearly, the graph is sub-linear. I applied different curve fitting models to this data to have an idea of the growth rate. The best fit was given by
$$ f_3(n) - f_1(n) \approx an^b $$
where $a \approx 0.1785$ and $b \approx 0.8122$. This experimental model has a very high $R^2 = 0.9999958$. If this is close to the true growth rate then, prime of the form $4k+1$ will always lag behind in the GPF race.
Question 2: What is the true growth rate of $f_3(n) - f_1(n)$?